Chapter 1 Approximation and Errors 1A p.2 Chapter 2 ...

1

Chapter 1 Approximation and Errors

1A p.2

1B p.13

Chapter 2 Manipulations and Factorization of Polynomials

2A p.28

2B p.36

2C p.50

2D p.60

2E p.69

2F p.78

Chapter 3 Identities

3A p.87

3B p.97

3C p.105

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2

F2A: Chapter 1A

Date Task Progress

Lesson Worksheet

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(Full Solution)

Book Example 1

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(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

3

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 1A Level 1


Teacher’s Signature

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___________ ( )




___________ ( )

Maths Corner Exercise 1A Multiple Choice

○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature

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E-Class Multiple Choice Self-Test

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_________

4

Book 2A Lesson Worksheet 1A (Refer to §1.1A–B)

1.1A Basic Concepts of Significant Figures

In a number, the non-zero digit with the largest place value is

called the first significant figure.

In general, all important digits of a number are called

significant figures.

e.g. The 1st

significant figure

Number of

significant figures

485 4 3

5 003 5 4

2.701 3 2 5

0.009 6 9 2

1. Complete the following table.

The 1st

significant figure

The 2nd

significant figure

Number of

significant figures

(a) 517

(b) 1 209

(c) 9.362

(d) 60.108

(e) 0.876

(f) 0.030 51

○○○○→→→→ Ex 1A 1, 2

1.1B Rounding off to the Required Significant Figures

Step 1111: Identify which digit is to be rounded.

Step 2222: Examine the next digit to the right. Then, round off

the number.

Step 3333: State the degree of accuracy.

e.g. Round off 1 593 to 2 significant figures.

1 593 = 1 600, cor. to 2 sig. fig.

4 8 5 The 3rd significant figure The 2nd significant figure The 1st significant figure

� Here, the two ‘0’s after ‘6’ are not significant figures. Step 3333

Step 1111: Step 2222:

1 5 9 3

1 6 0 0

1 5 9 3

� 0.009 6 The ‘0’s in this decimal are

place holders only. They are not significant figures. Note that the bold digits are significant figures.

�

The 2nd significant figure (i.e. the digit to be rounded)

5

Example 1 Instant Drill 1

Round off 64 257 to

(a) 2 significant figures,

(b) 3 significant figures,

(c) 4 significant figures.

Sol (a) 64 257

= 64 000, cor. to 2 sig. fig.

(b) 64 257

= 64 300, cor. to 3 sig. fig.

(c) 64 257

= 64 260, cor. to 4 sig. fig.

Round off 38 506 to




Sol (a) 38 506

= , cor. to

(b) 38 506

= ,

(c) 38 506

= ,

2. Round off 50 903 to

(a) 1 significant figure,



3. Round off 277 041 to




4. In the table below, round off each number to the required degree of accuracy.

Correct to

1 significant figure 2 significant figures 3 significant figures

(a) 1 048

(b) 69 463

(c) 859 032

○○○○→→→→ Ex 1A 3(a)(b), 4(a)(b), 5, 6

6 4 2 5 7

6 4 0 0 0

6 4 2 5 7

6 4 3 0 0

6 4 2 5 7

6 4 2 6 0

3 8 5 0 6

3 _________

3 8 5 0 6

3 8 _______

3 8 5 0 6

3 8 5 ____

In the answers, which of the ‘0’s (i) are significant figures?

(ii) are not significant figures?

6


Round off 0.170 3 to




Sol (a) 0.170 3

= 0.2, cor. to 1 sig. fig.

(b) 0.170 3

= 0.17, cor. to 2 sig. fig.

(c) 0.170 3

= 0.170, cor. to 3 sig. fig.

Round off 0.060 35 to




Sol (a) 0.060 35

= ( )

(b) 0.060 35

=

(c) 0.060 35

=

5. Round off 0.406 14 to




6. Round off 3.027 049 to




○○○○→→→→ Ex 1A 3(c)(d), 4(c)(d), 7–9

7. In the table below, round off each number to the required degree of accuracy.

Correct to

1 significant figure 2 significant figures 3 significant figures

(a) 2.955

(b) 73.014 7

(c) –135.71

○○○○→→→→ Ex 1A 10–12

0.1 7 0 3

0.2

0.1 7 0 3

0.1 7

0.1 7 0 3

0.1 7 0

0. 0 6 0 3 5

0. 0 6 _______

0. 0 6 0 3 5

0. 0 6 0 _____

0. 0 6 0 3 5

0. 0 _________

7


In each of the following, how many significant

figures does the number 800 have?

(a) 800, correct to the nearest hundred

(b) 800, correct to the nearest ten

(c) 800, correct to the nearest integer

Sol

(a) 800, cor. to the nearest hundred, has 1

significant figure.

(b) 800, cor. to the nearest ten, has 2


(c) 800, cor. to the nearest integer, has 3


In each of the following, how many significant

figures does the number 2 500 have?

(a) 2 500, correct to the nearest hundred

(b) 2 500, correct to the nearest ten

(c) 2 500, correct to the nearest integer

Sol (a) 2 500, cor. to the nearest hundred,

has .

(b) 2 500, cor. to the nearest ten, has

.

(c) 2 500, cor. to the nearest integer, has

.

8. How many significant figures are there in each of the following numbers?

(a) 4 520, correct to the nearest ten

(b) 78 000, correct to the nearest thousand

(c) 6 000 000, correct to the nearest hundred


(a) 2.90, correct to the nearest 0.01

(b) 0.07 0, correct to 3 decimal places

(c) 0.001 00, correct to 5 decimal places

○○○○→→→→ Ex 1A 13, 14


Oxygen will change into solid at –218.79°C.

Round off this temperature to


(b) 3 significant figures.

Sol (a) –218.79°C

= –220°C, cor. to 2 sig. fig.

(b) –218.79°C

= –219°C, cor. to 3 sig. fig.

The population of a city is 909 316. Round off

this population to



Sol (a) 909 316

= , ____________________

(b) 909 316

=

Similar to integers, the ‘0’s after the non-zero digits are significant figures.

significant figures

(a) 8 0 0 (b) 8 0 0 (c) 8 0 0 hundreds place tens place units place

8

10. A highway is 5.448 1 km long. Round off

this length to



11. A company’s bank balance was –$41 592 last

month. Round off this balance to



○○○○→→→→ Ex 1A 15, 16

�� ‘Explain Your Answer’ Question

12. An observatory records a rainfall of 3.602 5 mm in a certain day.

(a) Round off this amount of rainfall to

(i) 2 significant figures, (ii) 3 significant figures.

(b) Mary claims that the approximate value in (a)(i) has the same degree of accuracy as that

in (a)(ii). Do you agree? Explain your answer.

(a) (i) 3.602 5 mm =

(ii)

(b) Since the approximate value in ______ has more significant figures than that in ______ ,

the approximate value in ______ is more accurate.

∴ The claim is (agreed / disagreed).

�� Level Up Questions

13. Convert

7

1 into a decimal, correct to 3 significant figures.

14. The perimeter of a regular hexagon is 53.942 3 cm.

(a) Round off this perimeter to 2 significant figures. (b) Use the approximate value in (a) to estimate the length of each side of the regular

hexagon.

Remember to write down the reason.

9

1 Approximation and Errors

Level 1

1. Write down the first significant figure of each of the following numbers.

(a) 565 (b) 20 102

(c) 37.58 (d) 0.043 7


(a) 144 (b) 6 008

(c) 0.030 6 (d) 50.021

(e) 0.004 9 (f) 320.023

3. Round off the following numbers to 1 significant figure.

(a) 471 (b) 2 080

(c) 0.543 (d) 0.057 6

4. Round off the following numbers to 2 significant figures.

(a) 692 (b) 84 508

(c) 4.636 2 (d) 0.197 5

In each of the questions [Nos. 5–12] below, round off the number to




5. 25 086 6. 4 051 673 7. 9.034 8 8. 0.065 193

9. 0.732 95 10. 3.989 8 11. −18.062 12. −320.796

13. In each of the following, how many significant figures does the number 58 000 have?

(a) 58 000, correct to the nearest integer

(b) 58 000, correct to the nearest ten

(c) 58 000, correct to the nearest hundred

(d) 58 000, correct to the nearest thousand

Consolidation Exercise 1A ��

10


(a) 32.0, correct to the nearest 0.1

(b) 40.08, correct to 2 decimal places

(c) 0.106, correct to 3 decimal places

(d) 0.007 00, correct to 5 decimal places

15. It is known that mercury changes into solid state at –38.829°C. Round off this temperature to



16. There are two buildings A and B. The height of building A is 235 m. The height of building B is 88 m

higher than that of building A. Round off their heights to 2 significant figures.

17. The perimeter of a square is 46.07 cm. Find the length of each side of the square, correct to



18. It is known that the distance travelled by a butterfly in a day is 46 033 cm. Convert this distance to

(a) m,

(b) km.

(Give the answers correct to 3 significant figures.)

Level 2

19. State all possible number(s) of significant figures in each of the following approximate values.

(a) 0.054 (b) 620

(c) 3 000 (d) 0.085 0

20. Convert each of the following fractions into a decimal, correct to 4 significant figures.

(a) 7

1− (b)

16

930

(c) 18

25 (d)

111

13

21. Evaluate 200 −

200

3 and give the answer correct to



11

22. In each of the following expressions, round off each number involved to 3 significant figures. Then,

estimate the value of the expression.

(a) 37.82 + 62.15 (b) 475.36 − 123.78

23. In each of the following expressions, round off each number involved to 1 significant figure. Then,


(a) 5 974 × 3.18 (b) 425 ÷ 7.83

24. In the expression 210.53 × 3.45 ÷ 6.97, round off each number involved to 2 significant figures. Then,


25. In the expression 680.3 ÷ 19.95 + 349.8, round off each number involved to 3 significant figures.

Then, estimate the value of the expression.

26. After rounding off an integer N to 2 significant figures, the approximate value 150 is obtained. Write

down the greatest possible value of N.

27. An aeroplane takes 4 hours and 35 minutes to fly from Hong Kong to Tokyo. Convert this time to

hours, correct to 3 significant figures.

28. In a sailing race, the finishing time of the champion team is 8 days 15 hours 48 minutes and

20 seconds. Convert this time to

(a) days, correct to 3 significant figures,

(b) minutes, correct to 4 significant figures.

29. The following table shows the monthly expenditure of Carol in the second half year of 2015.

Month Jul Aug Sep Oct Nov Dec

Expenditure ($) 11 450 12 074 10 586 9 710 10 365 15 248

Find the total expenditure of Carol in the second half year of 2015.

(Give the answer correct to 4 significant figures.)

30. A construction company plans to replace all the tiles in a shopping mall. The total floor area of the

shopping mall is 75 120 m2 and the total cost of the tiles required is $47 980 000.

(a) Round off the total floor area and the total cost to 1 significant figure. Then, estimate the average

cost of each m2 of tile.

(b) Round off the total floor area and the total cost to 2 significant figures. Then, estimate the

average cost of each m2 of tile.

(c) Which estimation is more accurate, the one in (a) or the one in (b)? Explain your answer.

12

Consolidation Exercise 1A (Answer)

1. (a) 5 (b) 2

(c) 3 (d) 4

2. (a) 3 (b) 4 (c) 3

(d) 5 (e) 2 (f) 6

3. (a) 500 (b) 2 000

(c) 0.5 (d) 0.06

4. (a) 690 (b) 85 000

(c) 4.6 (d) 0.20

5. (a) 25 000 (b) 25 100 (c) 25 090

6. (a) 4 100 000 (b) 4 050 000

(c) 4 052 000

7. (a) 9.0 (b) 9.03 (c) 9.035

8. (a) 0.065 (b) 0.065 2 (c) 0.065 19

9. (a) 0.73 (b) 0.733 (c) 0.733 0

10. (a) 4.0 (b) 3.99 (c) 3.990

11. (a) −18 (b) −18.1 (c) −18.06

12. (a) −320 (b) −321 (c) −320.8

13. (a) 5 (b) 4

(c) 3 (d) 2

14. (a) 3 (b) 4

(c) 3 (d) 3

15. (a) −39°C (b) −38.8°C

16. building A: 240 m, building B: 320 m

17. (a) 11.5 cm (b) 11.52 cm

18. (a) 460 m (b) 0.460 km

19. (a) 2 (b) 2 or 3

(c) 1, 2, 3 or 4 (d) 3

20. (a) −0.142 9 (b) 30.56

(c) 1.389 (d) 0.117 1

21. (a) 200 (b) 200.0

22. (a) 100 (b) 351

23. (a) 18 000 (b) 50

24. 105

25. 384

26. 154

27. 4.58 h

28. (a) 8.66 days (b) 12 470 min

29. $69 430

30. (a) $625 (b) $640 (c) (b)

13

F2A: Chapter 1B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

14



(Full Solution)

Maths Corner Exercise 1B Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 1B Multiple Choice


___________ ( )



_________

15

Book 2A Lesson Worksheet 1B (Refer to §1.2)

1.2A Nature of Approximation in Measurement

The result obtained in measurement is not exact. It is only an approximate value.

1.2B Errors

I. Absolute Error

In estimation or measurement, the difference between the actual

value (i.e. exact value) and the approximate value is called the

absolute error.

Case 1111: If the actual value > the approximate value, then �

absolute error = actual value – approximate value

Case 2222: If the approximate value > the actual value, then �

absolute error = approximate value – actual value

Note: Absolute error is always positive.


Find the absolute errors in the following cases.

actual value approximate value

(a) 338 320

(b) 4.37 g 5 g

Sol (a) Absolute error = 338 – 320 � Case 1111

= 18

(b) Absolute error = (5 – 4.37) g � Case 2222

= 0.63 g

Find the absolute errors in the following cases.

actual value approximate value

(a) 1 891 2 000

(b) 0.905 s 0.9 s

Sol (a) Absolute error = ( ) – ( )

=

(b) Absolute error = [( ) – ( )] s

=


(a)

Actual temperature = 26.25°C

(b)

Actual height = 16.38 cm

(c)

Actual time = 44.691 s

Measured

value

Absolute

error

[( ) – ( )]°C

=

○○○○→→→→ Ex 1B 1–3

absolute error

16


Suppose the actual weight of a box of candy is

0.85 kg. Tom and John measure the weight of

the box of candy as 0.8 kg and 1 kg

respectively.

(a) Find the absolute errors of their

measurements.

(b) Whose measurement is more accurate?

Explain your answer.

Sol (a) Absolute error of Tom’s measurement

= (0.85 – 0.8) kg

= 0.05 kg

Absolute error of John’s measurement

= (1 – 0.85) kg

= 0.15 kg

(b) ∵ 0.05 kg < 0.15 kg

i.e. The absolute error of Tom’s

measurement is less than that of

John’s measurement.

∴ Tom’s measurement is more

accurate.

Suppose the actual temperature of a classroom

is 26.49°C. Amy and May measure the

temperature of the classroom as 26°C and

26.5°C respectively.


measurements.

(b) Whose measurement is more accurate?


Sol (a) Absolute error of Amy’s measurement

= [( ) – ( )]°C

=

Absolute error of May’s measurement

= [( ) – ( )]°C

=

(b) ∵ ( )°C < ( )°C

i.e.

2. Suppose there are 473 biscuits in a can.

Sam and Paul estimate the number of

biscuits in the can as 450 and 500

respectively.


estimations.

(b) Whose estimation is more accurate?


3. The price of a vase is $3 382. Mandy and

Sandy estimate the price of the vase as

$3 200 and $3 400 respectively.


estimations.

(b) Whose estimation is more accurate?


○○○○→→→→ Ex 1B 4, 5

17

II. Maximum Error

Maximum absolute error (or maximum error) =

2

1 × scale interval on the measuring tool

e.g. Use the ruler on the right in measurement.

Maximum error = 12

1× cm = 0.5 cm


Measuring tool Scale interval Maximum error

(a) Ruler 1 mm

(b) Measuring cylinder 10 mL

(c) Mechanical scale 0.4 kg

Lower limit of the actual value = measured value – maximum error

Upper limit of the actual value = measured value + maximum error

The possible range of the actual value is:

lower limit ≤ actual value < upper limit


The scale interval on a ruler is 1 cm. The

length of a pencil is measured to be 10 cm by

the ruler.

(a) Find the maximum error of the

measurement.

(b) If the actual length of the pencil is x cm,

find the possible range of x.

Sol (a) Maximum error =

2

1 × 1 cm

= 0.5 cm

(b)

Lower limit of the actual length

= (10 – 0.5) cm

= 9.5 cm

Upper limit of the actual length

= (10 + 0.5) cm

= 10.5 cm

∴ The possible range of x is

9.5 ≤ x < 10.5.

The scale interval on a stopwatch is 0.1 s.

Using this stopwatch, the measured time for

Ada to finish a swimming event is 27.2 s.

(a) Find the maximum error of the

measurement.

(b) If the actual time is a s, find the possible

range of a.


2

1 × ( ) s

=

(b)

Lower limit of the actual time

= [( ) – ( )] s

=

Upper limit of the actual time

= [( ) + ( )] s

=

∴ The possible range of a is

.

scale interval = 1 cm

Can x be 10.5?

11 cm 9 cm

upper limit

0.5 cm

possible range of the actual length

lower limit

0.5 cm

10 cm (measured value)

( ) s

upper limit

( ) s

possible range of the actual time

lower limit

27.2 s (measured value)

( ) s

( ) s

�‘≤’ means ‘is less than or equal to’.

18

5. In the figure, the weight of

the dumb-bell is measured to

be 1.6 kg.

(a) Find the maximum error

of the measurement.

(b) If the actual weight of the dumb-bell

is y kg, find the possible range of y.

(a) Scale interval of the weighing device

=

∴ Maximum error =

(b)

6. Samuel uses a measuring cup with a scale

interval of 50 mL to measure the volume of

a bottle of lemon tea. The result is 500 mL.

Find the lower limit and the upper limit of

the actual volume of the lemon tea.

7. The weight of a letter is measured as 8.12 g,

correct to the nearest 0.02 g.

(a) Find the maximum error of this

measured weight.

(b) If the actual weight of the letter is x g,

find the possible range of x.

8. The thickness of a book is measured as

38 mm, correct to the nearest mm. Find the

lower limit and the upper limit of the

actual thickness of the book.

○○○○→→→→ Ex 1B 6–8

‘Correct to the nearest 0.02 g’ can be regarded as the measuring tool with scale interval 0.02 g.

19

1.2C Relative Error

In measurement,

relative error =

valuemeasured

error maximum


The length of Shing Mun River in Sha Tin is

7 km, correct to the nearest km. For this

measured length, find

(a) the maximum error,

(b) the relative error.


2

1 × 1 km

= 0.5 km

(b) Relative error =

km 7

km 5.0

=

14

1

The weight of a smartphone is 120 g, correct to

the nearest 10 g. For this measured weight, find




2

1 × ( ) g

=


g ) (

g ) (

=

9. The volume of a cup is 250 mL, correct to

the nearest 5 mL. Find the relative error.

10. The time spent by Sunny to run along a

path is 22.5 s, correct to the nearest 0.5 s.

Find the relative error.

○○○○→→→→ Ex 1B 9, 10

Relative error has no units.

� Maximum error

=

2

1 × scale interval on the measuring tool

20

In estimation,

relative error =

valueactual

error absolute


There are 2 500 audience in a magic show. Neo

estimates the number of audience as 2 600.

Find, in Neo’s estimation,

(a) the absolute error,


Sol (a) Absolute error = 2 600 – 2 500

= 100


500 2

100

= 0.04

A chef estimates that there are 42 dishes in a

kitchen cabinet. If the actual number of dishes

in the kitchen cabinet is 60, find, in the chef’s

estimation,



Sol (a) Absolute error = ( ) – ( )

=


) (

) (

=

11. Kelly estimates that there are 43 floors in a

building. If there are only 40 floors in the

building, find the relative error of her

estimation.

12. The price of a necklace is $13 800. Mr

Chan estimates the price of the necklace as

$12 000. Find the relative error of his

estimation, correct to 2 significant figures.

○○○○→→→→ Ex 1B 11

� An absolute error is the difference between the actual value and the approximate value.

21

1.2D Percentage Error

Percentage error = relative error × 100%


The length of a wire is 5.0 cm, correct to the

nearest 0.5 cm. Find the percentage error of

this length.

Sol Maximum error =

2

1 × 0.5 cm

= 0.25 cm

Percentage error =

cm 0.5

cm 25.0 × 100%

= 5%

Sammy estimates that there are 600 words in

an article. If the actual number is 500, find the

percentage error of her estimation.

Sol Absolute error = ( ) – ( )

=

Percentage error =

) (

) ( × 100%

=

13. An engineer estimates that there are

28 000 seats in a stadium. If the actual

number of seats is 27 950, find the

percentage error, correct to 3 significant

figures.

14. The weight of a box of washing powder is

(500 ± 10) g. Find the percentage error.

15. It is known that the height of a lighthouse

is (32 ± 0.2) m. Find the percentage error.

16. The running time of a film is

(186 ± 6) minutes. Find the percentage

error, correct to the nearest 0.1%.

○○○○→→→→ Ex 1B 12, 13

Relative error = valuemeasured

error maximum Relative error = valueactual

error absolute (500 – 10) g ≤ actual value < (500 + 10) g, i.e. maximum error = 10 g

22

�� ‘Explain Your Answer’ Questions

17. The weight of a packet of potato chips is measured as 42 g, correct to the nearest g.

(a) Find the maximum error of the measurement.

(b) Can the actual weight of the packet of potato chips be less than 40 g? Explain your

answer.

(a) Maximum error =

(b) Lower limit of the actual weight =

∵ ________ g (> / = / <) 40 g

∴ The actual weight of the packet of potato chips (can / cannot) be less than 40 g.

18. The height of James is 160 cm, correct to the nearest 5 cm. If the actual height of Nick is

163 cm, is it possible that James is taller than Nick? Explain your answer.

�� Level Up Question

19. It is known that the monthly salary of Miss Cheung is $24 730. After rounding off her monthly

salary to 2 significant figures, find


(b) the percentage error.

(Give the answers correct to 3 significant figures if necessary.)

Compare 40 g with the lower limit of the actual weight.

Find the approximate value of her

monthly salary first.

23

1 Approximation and Errors

Level 1

1. In the figure, the size of ∠AOB is measured by

a protractor.

(a) Find the size of ∠AOB, correct to the

nearest 1°.

(b) If the actual size of ∠AOB is 110.2°,

find the absolute error of the approximate

value in (a).

2. In the figure, the length of a drumstick is measured by the given

ruler.

(a) Find the length of the drumstick, correct to the nearest 1 cm.

(b) If the actual length is 38.97 cm, find the absolute error of the

approximate value in (a).

3. The figure shows a reading on a mechanical scale.

(a) Find the reading, correct to the nearest 0.5 kg.

(b) If the actual weight is 5.38 kg, find the absolute error of the

approximate value in (a).

4. Suppose the actual volume of a bottle of milk is 285.3 mL. Judy and Flora measure the volume of the

milk as 285.5 mL and 286 mL respectively.

(a) Find the absolute errors of their measurements.

(b) Whose measurement is more accurate? Explain your answer.

5. There are 683 guests in a party. Aaron and Larry estimate the number of guests as 690 and 680

respectively.

(a) Find the absolute errors of their estimations.

(b) Whose estimation is more accurate? Explain your answer.


Consolidation Exercise 1B ��

cm0 5 10 15 20 25 30 35 40

30 35 40

24

Measuring tool Scale interval Maximum error

(a) Timer 1 s

(b) Syringe 0.2 mL

(c) Measuring tape 5 mm


Measured value Maximum error Lower limit of

actual value

Upper limit of

actual value

(a) 57 cm 56 cm

(b) 24°C 0.1°C

(c) 500 mL 500.5 mL

(d) 0.05 kg 77.80 kg

8. In a zoo, the weight of an elephant is measured as 4 550 kg, correct to the nearest 10 kg.

(a) Find the maximum error of the measured weight.

(b) If the actual weight of the elephant is x kg, find the possible range of x.

9. The figure shows a clinical thermometer.

(a) Find the scale interval on this thermometer.

(b) Using this thermometer, the body temperature of a man

is measured as 36.3°C. If his actual body temperature

is x°C, find the possible range of x.

10. The area of a park is 435 m2, correct to the nearest m2. For this measured area, find



11. The length of a bridge is measured as 2 500 m, correct to the nearest 10 m. Find its relative error.

12. Mandy estimates that there are 234 words on a certain page of a book. If the actual number of words

on this page is 240, find the relative error of Mandy’s estimation.

13. The volume of water in a bathtub is (680 ± 5) L. Find the percentage error of this measurement,

correct to 2 significant figures.

14. The speed of a motorcycle is measured by the speedometer shown on the

right. Find the percentage error of the measured speed, correct to the

nearest 0.1%.

35 37 39 41 °C

36 38 40 42

25

Level 2

15. There were 5 648 vehicles passing through a tunnel yesterday.

(a) Find the absolute error after rounding off the number of vehicles to

(i) 2 significant figures, (ii) 3 significant figures.

(b) Which approximate value obtained in (a) is more accurate? Explain your answer.

16. The figure shows a sticker in a triangular shape. The base and the height

are measured as 65 mm and 54 mm respectively. Both measurements

are correct to the nearest mm.

(a) Find the maximum error of each measurement.

(b) Find the minimum possible area of the sticker.

(c) If the actual area of the sticker is y mm2, find the possible range of y.

17.

0 1

0

cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 3

In the polygon as shown, all the lengths are measured by the given ruler.

(a) Find the maximum error of each measurement.

(b) Find the minimum possible area of the polygon.

(c) If the actual area of the polygon is x cm2, find the possible range of x.

18. (a) The weight of a doll is measured as 54 g, correct to the nearest g. Find the minimum possible

weight of the doll.

(b) The weight of a robot is measured as 4.5 kg, correct to the nearest 0.5 kg.

(i) Can the actual weight of the robot exceed 4 760 g? Explain your answer.

(ii) Is it possible that the actual weight of the robot is equal to 90 times the actual weight of the

doll in (a)? Explain your answer.

19. Ben measures the weight of 800 identical clips as 640 g and the relative error is

160

1. If the

actual weight of each clip is x g, find the possible range of x.

20. Mrs Chau estimates the price of an oil painting. The absolute error and the relative error of Mrs

Chau’s estimation are $3 780 and

95

18 respectively. Is the price of the oil painting greater than

$20 000? Explain your answer.

21. Last week, 36 924 people visited a theme park. After rounding off the number of people to

2 significant figures, find


(b) the percentage error, correct to the nearest 0.1%,

�

26

of the approximate value.

22. Gloria uses a measuring cylinder to measure the volume of

some water. The result is shown on the right.

(a) What is the reading on the measuring cylinder?

(b) What is the maximum error of this reading?

(c) What is the percentage error of this reading?

(Give the answer correct to 2 significant figures.)

23. Two thermometers P and Q are used to measure the temperature in a laboratory. The scale interval of

thermometer P is 2°C and the scale interval of thermometer Q is 0.1°C. Assume the actual

temperature in the laboratory is 28.385°C. Complete the following table. (Give the answers correct to

3 significant figures if necessary.)

Measured

value ( C° )

Absolute

error ( C° )

Relative

error

Percentage

error

(a) Thermometer P

(b) Thermometer Q

24. The volume of a bottle of oil is measured as 3.00 L, correct to the nearest 0.01 L. The volume of a can

of coffee is measured as 260 mL, correct to the nearest mL.

(a) Find the percentage errors of these two measured values.

(Give the answers correct to 3 significant figures.)

(b) Which measured value is more accurate? Explain your answer.

25. The measured weight of a bag of rice is 12 kg. The percentage error of this measurement is

3

18 %.


(b) Is it possible that the actual weight of the bag of rice is 10.8 kg? Explain your answer.

26. The International Commerce Centre is the tallest building in Hong Kong. The measured height of the

building is 475 m with percentage error 2%.


(b) Is it possible that the actual height of the building is 485 m? Explain your answer.

27

Consolidation Exercise 1B (Answer)

1. (a) 110° (b) 0.2°

2. (a) 39 cm (b) 0.03 cm

3. (a) 5.5 kg (b) 0.12 kg

4. (a) Judy: 0.2 mL, Flora: 0.7 mL

(b) Judy’s measurement

5. (a) Aaron: 7, Larry: 3

(b) Larry’s estimation

6. (a) 0.5 s (b) 0.1 mL (c) 2.5 mm

7. (a) maximum error: 1 cm, upper limit: 58

cm

(b) lower limit: 23.9°C, upper limit: 24.1°C

(c) maximum error: 0.5 mL, lower limit: 499.5

mL

(d) measured value: 77.85 kg, upper limit: 77.90

kg

8. (a) 5 kg (b) 4 545 ≤ x < 4 555

9. (a) 0.1°C (b) 36.25 ≤ x < 36.35

10. (a) 0.5 m2 (b) 870

1

11. 500

1

12. 0.025

13. 0.74%

14. 1.3%

15. (a) (i) 48 (ii) 2

(b) (a)(ii)

16. (a) 0.5 mm (b) 1 725.375 mm2

(c) 1 725.375 ≤ y < 1 784.875

17. (a) 0.05 cm (b) 15.56 cm2

(c) 15.56 ≤ x < 16.57

18. (a) 53.5 g

(b) (i) no (ii) no

19. 0.795 ≤ x < 0.805

20. no

21. (a) 76 (b) 0.2%

22. (a) 42 mL (b) 1 mL (c) 2.4%

23. Measured value ( C° )

Absolute error ( C° )

Relative error

Percentage error

(a) 28 0.385 0.013 6 1.36%

(b) 28.4 0.015 0.000 528 0.052 8%

24. (a) oil: 0.167%, coffee: 0.192%

(b) oil

25. (a) 1 kg (b) no

26. (a) 9.5 m (b) no

28

F2A: Chapter 2A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

29

Book 2A Lesson Worksheet 2A (Refer to §2.1)

2.1 Index Notation

(a) For any number a and any positive integer n,

a × a × a × … × a = an.

(b) an is read as ‘a to the power n’ or ‘the nth power of a’.


Use index notation to represent each of the

following.

(a) the square of 8

(b) 10 to the power 6

(c) the 11th power of x

Sol (a) The square of 8 = 82

(b) 10 to the power 6 = 106

(c) The 11th power of x = x11


following.

(a) the cube of 7

(b) 5 to the power 8

(c) the 23rd power of y

Sol (a) The cube of 7 =

(b) 5 to the power 8 =

(c) The 23rd power of y =

○○○○→→→→ Ex 2A 1, 2



following.

(a) 9 × 9 × 9 × 9

(b) 2 × 2 × 2 × 3 × 3 × 3 × 3

(c) (–6) × (–6) × (–6) × (–6) × (–6)

(d) 44

7777

×

×××

(e) a × a × a × a × a × a

(f) x ⋅ x ⋅ x ⋅ y ⋅ y

Sol (a) 9 × 9 × 9 × 9 = 94

(b) 2 × 2 × 2 × 3 × 3 × 3 × 3 = 23 × 34

(c) (–6) × (–6) × (–6) × (–6) × (–6) = (–6)5

(d) 44

7777

×

××× =

2

4

4

7

(e) a × a × a × a × a × a = a6

(f) x ⋅ x ⋅ x ⋅ y ⋅ y = x3y2


following.

(a) 4 × 4 × 4 × 4 × 4

(b) 7 × 7 × 7 × 6 × 6

(c) b × b × b

(d) m × m × m × m × n × n

(e) (–d) × (–d) × (–d)

(f) qqqqq

ppp

××××

××

Sol (a) 4 × 4 × 4 × 4 × 4 =

(b) 7 × 7 × 7 × 6 × 6 =

(c) b × b × b =

(d) m × m × m × m × n × n =

(e) (–d) × (–d) × (–d) =

(f) qqqqq

ppp

××××

××=

n times

� a is called the base and

n is called the index.

‘The square of 8’ means ‘8 to the power 2’.

‘The cube of 7’ means ‘7 to the power ’.

‘⋅’ means the same as ‘×’.

30

Use index notation to represent each of the following. [Nos. 1–12]

1. 5 × 5 × 5 × 5

2. 7 × 7 × 10 × 10 × 10 × 10 × 10

3. (–2) × (–2) × (–2) 4. –(6 × 6 × 6)

5. 8 × 8 × 8 × 8 × 9 × 9 6. 333333

1111

×××××

×

7. r × r × r × r × r × r × r 8. 2 × 2 × 2 × d × d × d × d × d

9. h × h × h × h × k 10. (–f ) × (–f ) × (–f ) × (–f ) × (–f )

11. –(a × a × a × a) 12. aaaa

bbb

⋅⋅⋅

−⋅−⋅− )()()(

○○○○→→→→ Ex 2A 3–9

31


Without using a calculator, find the values of

the following expressions.

(a) 24

(b) –34

(c) (–5)3

Sol (a) 24 = 2 × 2 × 2 × 2

= 16

(b) –34 = –(3 × 3 × 3 × 3)

= –81

(c) (–5)3 = (–5) × (–5) × (–5)

= –125



(a) 43

(b) –53

(c) (–2)5

Sol (a) 43 = ( ) × ( ) × ( )

=

(b) –53 = –( ___ × ___ × ___ )

=

(c) (–2)5 = (–2) × (–2) × ______________

=

Without using a calculator, find the values of the following expressions. [Nos. 13–18]

13. 35 14. (–5)4

15. 6(2)3 16. –8(–3)2

17. (a) 63 18. (a) 92

(b) 42

(c) 63 + 42

(b) 25

(c) 92 – 25

○○○○→→→→ Ex 2A 10–13, 16(a), (b)

32


19. In each of the following, are the two expressions equal in value? Explain your answer.

(a) (– 5)3, 53 (b) – 34, (– 3)4

(a) (– 5)3 = (b) – 34 =

53 =


20. Use index notation to represent each of the following.

(a) 3 × 3 × (–2) × (–2) × (–2)

(b) (– 4) × 7 × (– 4) × 7 × (– 4) × 7 × (– 4)

(c) m × (–n) × (–n) × (–n) × m

21. Given that R = 4s – u3, find the value of R in each of the following cases.

(a) s = 8, u = –2 (b) s = ,2

1u = 3

Put the same number together.

33

2 Manipulations and Factorization of Polynomials

Level 1

Use index notation to represent each of the following. [Nos. 1−−−−10]

1. (a) the square of 9 (b) the cube of m

2. (a) the 7th power of 4 (b) k to the power 8

3. (a) 5 × 5 × 5 (b) 7 × 7 × 7 × 7

4. (a) 3 × 3 × 3 × 2 (b) 4 × 4 × 4 × 4 × 4 × 6 × 6

5. (a) h × h (b) k × k × k × k × k × k × k

6. (a) m × p × p × p (b) n × n × n × n × n × n × q × q

7. (a) (−3) × (−3) × (−3) × (−3) × (−3) (b) −(10 × 10 × 10 × 10 × 10 × 10)

8. (a) (−r) × (−r) × (−r) × (−r) (b) −(s × s × s × s × s × s × s)

9. (a) 4 × 4 × c × c × c (b) 9 × 9 × 9 × 9 × d × d

10. (a) 333333

2222

×××××

××× (b)

aaaaa ××××

××× 8888

Without using a calculator, find the values of the following expressions. [Nos. 11−−−−14]

11. (a) 25 (b) (−3)2

12. (a) −43 (b) −(−5)3

13. (a) 6(24) (b) −2(−72)

14. (a) 3)10(

2

− (b)

4)3(

1

−


34

Level 2

Use index notation to represent each of the following. [Nos. 15−−−−16]

15. (a) (−4) × 7 × (−4) × 7 × 7

(b) (−1) × (−8) × (−1) × (−1) × (−1) × (−8)

16. (a) q × (−p) × (−p) × q × q × (−p) × q × q

(b) −(h × m × h × h × m × h × m)

Without using a calculator, find the values of the following expressions. [Nos. 17−−−−19]

17. (a) 72 − 23 (b) −62 + 52

(c) 34 − (−4)2 + 25

18. (a) (−1)7 × (−33) (b)

3

3

1

× 92

(c) 62 ×

4

2

1

− × (−1)6

19. (a) (−3)3 + 24 × 52 (b) 43 ×

5

2

1

− + 32

(c) (−1)5 −

3

6

1

− × 64

20. Given that P = m3 − n2, find the value of P in each of the following cases.

(a) m = 3, n = 4 (b) m = 1, n = 7 (c) m = −2, n = −1

21. Given that C = h2 + k5, find the value of C in each of the following cases.

(a) h = 4, k = 2 (b) h = 5, k = −2 (c) h =

3

1, k = −1

22. Given that D = −(p4 + 2q3), find the value of D in each of the following cases.

(a) p = −1, q = 2 (b) p = −2, q = −3 (c) p =

2

1− , q =

4

1

35


1. (a) 92 (b) m3

2. (a) 47 (b) k8

3. (a) 53 (b) 74

4. (a) 33 × 2 (b) 45 × 62

5. (a) h2 (b) k7

6. (a) mp3 (b) n6q2

7. (a) (−3)5 (b) −106

8. (a) (−r)4 (b) −s7

9. (a) 42c3 (b) 94d 2

10. (a) 6

4

3

2 (b)

5

48

a

11. (a) 32 (b) 9

12. (a) −64 (b) 125

13. (a) 96 (b) 98

14. (a) 500

1− (b)

81

1

15. (a) (−4)2 × 73 (b) (−1)4 × (−8)2

16. (a) (−p)3q5 (b) −h4m3

17. (a) 41 (b) −11 (c) 97

18. (a) 27 (b) 3 (c) 4

9

19. (a) 373 (b) 7 (c) 5

20. (a) 11 (b) −48 (c) −9

21. (a) 48 (b) −7 (c) 9

8−

22. (a) −17 (b) 38 (c) 32

3−

36

F2A: Chapter 2B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)




___________ ( )

Maths Corner Exercise ○ Complete and Checked Teacher’s

37

2B Level 2 ○ Problems encountered ○ Skipped

Signature ___________ ( )




___________ ( )



___________ ( )



_________

38


2.2A Multiplication of Powers

If m and n are positive integers, then

am × an = am + n


Simplify the following expressions and express

the answers in index notation.

(a) 82 × 83

(b) a5 × a

(c) y7 ⋅ y10

Sol (a) 82 × 83 = 82 + 3

= 85

(b) a5 × a = a5 + 1 � a = a1

= a6

(c) y7 ⋅ y10 = y7 + 10

= y17



(a) 52 × 56

(b) e × e8

(c) h5 ⋅ h13

Sol (a) 52 × 56 = 5( ) + ( )

=

(b) e × e8 = e( )

=

(c) h5 ⋅ h13 =

Simplify the following expressions and express the answers in index notation. [Nos. 1– 4]

1. 63 × 65

2. y4 × y2

3. x12 ⋅ x

4. r5 ⋅ r2 ⋅ r8

○○○○→→→→ Ex 2B 1, 5

Add the indices.

� Apply to the powers with

the SAME base only.

39


Simplify the following expressions.

(a) 5c3 × 2c9

(b) m2n × 7m11n4

Sol (a) 5c3 × 2c9

= 5 × 2 × c3 × c9

= 10c3 + 9

= 10c12

(b) m2n × 7m11n4

= 7 × m2 × m11 × n × n4

= 7m2 + 11n1 + 4

= 7m13n5


(a) 4w 8 × 10w

6

(b) a3b5 × (–3a2b4)

Sol (a) 4w 8 × 10w

6

= 4 × ( ) × w ( )

× w ( )

= ( ) × w ( )

=

(b) a3b5 × (–3a2b4)

= ( ) × a( ) × a( ) × ___________

=

Simplify the following expressions. [Nos. 5–8]

5. (–3a4) × 2a6

6. p2q5 × pq3

7. (–x6y10) × 5x9y3

8. (–8m4n7) × (– 4m5n13)

○○○○→→→→ Ex 2B 6, 13(a), 14

Multiply the number parts and put the variable parts together.

Put the powers with the same bases together.

40

2.2B Divison of Powers

If m and n are positive integers and a ≠ 0, then

am ÷ an = am – n (where m > n)

am ÷ an = mn

a−

1 (where m < n)




(a) 65 ÷ 62

(b) t8 ÷ t3

(c) 57 ÷ 513

(d) u6 ÷ u10

Sol (a) 65 ÷ 62 = 65 – 2

= 63

(b) t8 ÷ t3 = t8 – 3

= t5

(c) 57 ÷ 513 = 7135

1−

=65

1

(d) u6 ÷ u10 = 610

1−

u

= 4

1

u



(a) 79 ÷ 73

(b) a12 ÷ a8

(c) 24 ÷ 211

(d) b3 ÷ b15

Sol (a) 79 ÷ 73 = 7( ) – ( )

=

(b) a12 ÷ a8 =

(c) 24 ÷ 211 = )()(2

1−

=

(d) b3 ÷ b15 =

Simplify the following expressions and express the answers in index notation. [Nos. 9–12]

9. 4

6

3

3

10. r17 ÷ r7

11. 108 ÷ 1019

12. 21

15

t

t

○○○○→→→→ Ex 2B 2, 7, 13(b)

4 (< / >) 11

Subtract the indices.

7 < 13

41



(a) 36w8 ÷ 9w2

(b) 12x2y ÷ (– 4xy3)

Sol (a) 36w8 ÷ 9w2 = 2

8

9

36

w

w

=

9

36w8 – 2

= 4w6

(b) 12x2y ÷ (– 4xy3) = 3

2

4

12

xy

yx

−

= 13

123−

−

−y

x

= 2

3

y

x−


(a) 15d 8 ÷ 3d

4

(b) 63e5f

7 ÷ 7e12f 3

Sol (a) 15d 8 ÷ 3d

4 =

)(

15 8d

=

(b) 63e5f

7 ÷ 7e12f 3 =


13. (a) 18v9 ÷ 6v4

(b) 2

72

32

4

xy

yx−

14. (a) 6

3

8

24

x

x

(b) 25x8y12 ÷ 15x6y16

○○○○→→→→ Ex 2B 8, 9, 13(c), 15

42

2.2C Powers of Powers

If m and n are positive integers, then

(am)n = am × n




(a) (34)5

(b) (a6)2

Sol (a) (34)5 = 34 × 5

= 320

(b) (a6)2 = a6 × 2

= a12



(a) (53)2

(b) (y4)9

Sol (a) (53)2 = 53 × ( )

=

(b) (y4)9 = y( )

=

Simplify the following expressions and express the answers in index notation. [Nos. 15–20]

15. (117)4

16. (x2)7

17. (y8)5

18. – 4(y9)3

19. m2(m3)6

20. 14

84 )(

m

m

○○○○→→→→ Ex 2B 3, 10

43

2.2D Powers of Products and Powers of Quotients

If n is a positive integer, then

(ab)n = anbn



(a) (xy)7

(b) (2a)3

(c) (u2v4)6

Sol (a) (xy)7 = x7y7

(b) (2a)3 = 23a3

= 8a3

(c) (u2v4)6 = (u2)6(v4)6

= u2 × 6v4 × 6

= u12v24


(a) (bc)9

(b) (5k)2

(c) (r3s8)4

Sol (a) (bc)9 = b( )c( )

(b) (5k)2 =

(c) (r3s8)4 = ( )( )( )( )

=


21. (ab)12

22. (3x2)4

23. (h5k10)6

24. 3m × 5m

○○○○→→→→ Ex 2B 4(a), 11

anbn = (ab)n

44

If n is a positive integer, then n

b

a

=

n

n

b

a (where b ≠ 0)



(a)

10

k

h (b)

23

4

x

Sol (a)

10

k

h =

10

10

k

h

(b)

23

4

x =

2

23

4

)(x

=

16

2 3×x

=

16

6x


(a)

8

s

r (b)

3

2

4

3

c

a

Sol (a)

8

s

r =

) (

) (

s

r

(b)

3

2

4

3

c

a =

) (

) ( 4 ×a

=


25.

52

a

26.

2

9

7

u

t

27.

4

2

3

−

w

28.

3

5

64

y

x

○○○○→→→→ Ex 2B 4(b), 12

45

29. 5y9 ÷ (y2)4

30. 2

b

a × (ab)3

○○○○→→→→ Ex 2B 16–19



31. (a) a2 × 3a5 × (– 4a7)

(b) 24y9 ÷ 3y2 ÷ y

4

(c) (h2k)3 ×

26

h

k

32. (9x)(3y)

Convert the base of 9x to 3. [Hint: 9 = 32.]

46


Level 1

Calculate the following expressions and express the answers in index notation. [Nos. 1−−−−4]

1. (a) 37 × 32 (b) −64 × 64

2. (a) 58 ÷ 5 (b) 11

6

2

2

3. (a) (74)5 (b) (106)3

4. (a) 4

4

2

8

−

− (b) (−6)9 × (−2)9

Simplify the following expressions. [Nos. 5−−−−18]

5. (a) p2 × p3 (b) h4 × h3 (c) m3 × (−m8)

6. (a) 2t6 × 3t4 (b) 4x5 × 3x2 (c) −2y7 × (−5y)

7. (a) 6

8

u

u (b) v3 ÷ v7 (c) w2 ÷ (−w5)

8. (a) x5 ÷ 2x3 (b) 6

2

3

6

y

y (c)

8

11

)(9

)(12

z

z

−

−

9. (a) (h5)3 (b) (k2)6 (c) [(−r)4]5

10. (a) (mn)9 (b) (−3a)3 (c) (pq2)4

11. (a)

7

d

c (b)

4

2

−

h (c)

3

5

3

4

n

m

12. (a) t4 × 3t2 × 5t5 (b) u9 ÷ u3 ÷ u4 (c) 20v12 ÷ v3 ÷ 4v6

13. (a) xy3 × x3y4 (b) 4w2z5 × 6w7z3 (c) (−9a5b2) × (−3a3b4)

14. (a) h7k5 ÷ h4k3 (b) 35p4q11 ÷ 7pq5 (c) 18r4s5 ÷ (−6r2s)

15. a7 × (−3a3) ÷ (−a5)


47

16. 16b14 ÷ (b2)5

17. h5 ÷ (gh3)2

18. m2 ×

3

m

n

In each of the following, a and b are positive integers. Write down two sets of values for a and b such that

the equality holds. [Nos. 19−−−−23]

19. 7a × 7b = 75

20. 5a ÷ 5b = 53

21. (4a)b = 418

22. a3 ⋅ b3 = 123

23. 7

7

b

a = 57

Level 2


24. (a) −35r6s3 ÷ 5r4s11 (b) 52

10

42

16

tt

t

×

25. (a) (3m)(27n) (b) q

p

10

100 (where p > q)

26. (a) 8a ÷ 4a (b) 3 × 9b

27. (a) (x2 × 2x)5 (b)

3

4

23

−

×

h

gg

28. (a) 532

523

)(

)(

qp

qp (b) (8a3b2)7 ÷ (−8a3b)5

29. (a) 23

4

)3()2(

36

xy

x

−× (b) 16h2 ×

3

24

3

k

h

30. m

mn

25

5125 × (where n > m)

48

31. (32n × 27n)3

32. (−d )3 × (−6d 3)2 ÷ 9d

4

33. (−m3n)2 ÷ m5n3 × mn4

34. (2x5y3)4 ÷ 32

423

)3(

)6(

xy

yx

−

35. 20a4b5 ÷ (−5b3) ÷ (−4a2b3)2

36. It is given that pm = 81 and pn = 27, where m, n are positive integers and m > n. Find the value of pm − n.

37. Given that ar = 4, where r is a positive integer, find the value of a3r.

38. Given that xh = 2 and xk = 8, where h and k are positive integers, find the value of x3h + k.

39. Given that xk = 5 and y

k = 2, where k is a positive integer, find the value of

2

1(xy2)k.

49


1. (a) 39 (b) −68

2. (a) 57 (b) 52

1

3. (a) 720 (b) 1018

4. (a) 44 (b) 129

5. (a) p5 (b) h7 (c) −m11

6. (a) 6t10 (b) 12x7 (c) 10y8

7. (a) u2 (b) 4

1

v (c)

3

1

w−

8. (a) 2

2x

(b) 4

2

y (c)

3

4 3z

−

9. (a) h15 (b) k12 (c) r20

10. (a) m9n9 (b) −27a3 (c) p4q8

11. (a) 7

7

d

c (b)

16

4h

(c) 15

9

64n

m

12. (a) 15t11 (b) u2 (c) 5v3

13. (a) x4y7 (b) 24w9z8 (c) 27a8b6

14. (a) h3k2 (b) 5p3q6 (c) −3r2s4

15. 3a5

16. 16b4

17. hg

2

1

18. m

n3

19. a = 1, b = 4; a = 2, b = 3

(or other reasonable answers)

20. a = 4, b = 1; a = 5, b = 2


21. a = 2, b = 9; a = 3, b = 6


22. a = 2, b = 6; a = 3, b = 4


23. a = 10, b = 2; a = 15, b = 3


24. (a) 8

27

s

r− (b) 2t3

25. (a) 3m + 3n (b) 102p − q

26. (a) 2a (b) 31 + 2b

27. (a) 32x15 (b) 12

927

h

g−

28. (a) 5

5

q

p (b) −64a6b9

29. (a) 3

2

2y

x (b)

6

5

4

27

k

h

30. 53n − m

31. 315n

32. −4d 5

33. m2n3

34. 3

1011yx

−

35. 44

1

b−

36. 3

37. 64

38. 64

39. 10

50

F2A: Chapter 2C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)



(Full Solution)

Maths Corner Exercise 2C Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 2C Multiple Choice


___________ ( )



_________

51

Book 2A Lesson Worksheet 2C (Refer to §2.3)

2.3A Basic Knowledge about Polynomials

(a) A monomial is an algebraic expression that consists of one single term.

This term must be one of the following types:

Note: x

1 and

234

2

y− are not monomials.

(b) A polynomial can be a monomial or a sum of monomials.

e.g. 3x, 2x – 1, 3x2 + 5xy

1. Determine whether each of the following algebraic expressions is a polynomial.

(a) –5 _____ (b) 7x _____ (c) 6x2y5 _____

(d) z

xy _____ (e)

8

2 3p

_____ (f) yx 53

4+ _____

(g) cb

aa 4

2

35 −+ _____ (h)

d

cba 3

5

2

2

23

−+ _____

○○○○→→→→ Ex 2C 1

(a) In each term of a polynomial, a coefficient is

the numerical value that is multiplied with the

variable(s).

(b) A constant term is the term without variables.

e.g. (i) The coefficient of the monomial – 4x2 is – 4.

(ii) For the polynomial 5x2 – 7x + 1:

Coefficient of the x2 term = 5

Coefficient of the x term = –7

Constant term = 1

Monomial Examples

(i) A number 3, – 4,

2

1

(ii) A variable with a positive integral

index x, y2

(iii)

The product of a number and one or

more variables, where each variable has

a positive integral index

6x, –7x2y3

� The polynomial 2x – 1 has two terms:

2x and –1.

coefficient of the x2 term = 5

coefficient of the x

term = –7

constant

term = 1

� 5x2 – 7x + 1

Can you explain why some of the expressions here are not polynomials?

52

2. Complete the table below.

Polynomial Number of terms

Coefficient of

the x term Constant term

(a) 9x + 4

(b) 3x2 – 5x – 11

(c) –7x3 + 8x – 6x2

2.3B Degree of a Polynomial

I. Monomials

Degree of a monomial = sum of the indices of all the variables

3. Write down the degree of each of the following monomials.

(a) 2

1 (b) 3x (c) 2x5

(d) –xy (e) x2y (f) 8x3y4

(g) 4

xyz (h) a2bc3 (i) – 4xyz5

(a) Degree of

2

1 =

(b) Degree of 3x =

(c) Degree of 2x5 =

(d) Degree of –xy = ( ) + ( )

=

(e) Degree of x2y = ( ) + ( )

=

(f) Degree of 8x3y4 = ( ) + ( )

=

(g) Degree of

4

xyz

= ( ) + ( ) + ( )

=

(h) Degree of a2bc3

=

(i) Degree of – 4xyz5

=

○○○○→→→→ Ex 2C 2

e.g. Monomial 3 x y2 x2y3 –5x2y4z6

Degree 0 1 2 2 + 3 = 5 2 + 4 + 6 = 12

When the monomial is a number, degree = 0.

53

II. Polynomials

Degree of a polynomial = degree of the term with the highest degree

e.g. Consider the polynomial –y3 + 5y2 + 3y – 4.

∴ The degree of –y3 + 5y2 + 3y – 4 is 3.


Write down the degree of each of the following

polynomials.

(a) 8 + 5x – 3x2

(b) 9x3y + x5y2

Sol (a)

∵ The term with the highest degree

is –3x2 and its degree is 2.

∴ The degree of 8 + 5x – 3x2 is 2.

(b)


is x5y2 and its degree is 7.

∴ The degree of 9x3y + x5y2 is 7.

Write down the degree of each of the following

polynomials.

(a) 4x2 – 8x

(b) y3 – 5x2y + 2xy3

Sol (a)


is ________ and its degree is ___.

∴ The degree of 4x2 – 8x is .

(b)


is ________ and its degree is ___.

∴ The degree of y3 – 5x2y + 2xy3

is .

Write down the degree of each of the following polynomials. [Nos. 4–7]

4. x5 – 6x3 + 7x2 – 1

5. y – 5y4 + 3y2

6. mn2 + 4m7n – 5

7. 8x6 – 5x4y3 + x – 9

○○○○→→→→ Ex 2C 3–7

Term –y3 +5y2 +3y – 4

Degree 3 2 1 0

8 + 5x – 3x2

Degree: 0 1 2

9x3y + x5y2

Degree: 3 + 1 5 + 2 = 4 = 7

4x2 – 8x

Degree: ___ ___

y3 – 5x2y + 2xy3

Degree:

highest degree

54

2.3C Arrangement of Terms of a Polynomial

Consider the polynomials x3 + x2 + x + 1 and 1 + y + y2 + y3.

1. The terms of x3 + x2 + x + 1 are arranged in descending powers of x.

2. The terms of 1 + y + y2 + y3 are arranged in ascending powers of y.


Arrange the terms of each of the following

polynomials in descending powers of x.

(a) 5x + 9 – 7x2

(b) 8x2 + 7x + 4x3 – 5

Sol (a) Arrange in descending powers of x:

–7x2 + 5x + 9

(b) Arrange in descending powers of x:

4x3 + 8x2 + 7x – 5


polynomials in descending powers of x.

(a) –9x + 3x2 – 8

(b) 4x3 – 2x5 – 6x + 1

Sol (a) Arrange in descending powers of x:

____________________________

(b) Arrange in descending powers of x:

____________________________



polynomials in ascending powers of y.

(a) – 4y + 9y2 – 10

(b) 6y2 – 2y3 + 1

Sol (a) Arrange in ascending powers of y:

–10 – 4y + 9y2

(b) Arrange in ascending powers of y:

1 + 6y2 – 2y3


polynomials in ascending powers of y.

(a) –y – 6 + 2y2

(b) 7y6 + 2y3 + 4 – 10y2

Sol (a) Arrange in ascending powers of y:

____________________________

(b) Arrange in ascending powers of y:

____________________________

For each of the polynomials in Nos. 8–9, arrange the terms in

(a) descending powers of x,

(b) ascending powers of x.

8. 8x + 10x2 + x4

9. 5x3 – x – 4 – 2x6

○○○○→→→→ Ex 2C 8, 9

Powers of x are decreasing.

Powers of y

are increasing.

55

2.3D Value of a Polynomial


Find the value of the polynomial 5x2 – 8x + 6

in each of the following cases.

(a) x = 3

(b) x = – 4

Sol (a) When x = 3,

the value of the polynomial

= 5(3)2 – 8(3) + 6

= 45 – 24 + 6

= 27

(b) When x = – 4,


= 5(– 4)2 – 8(– 4) + 6

= 80 + 32 + 6

= 118

Find the value of the polynomial –x3 + 4x2 in

each of the following cases.

(a) x = 2

(b) x = –3

Sol (a) When x = ___,


= –( )3 + 4( )2

=

(b) When x = _____,


=

10. Find the value of the polynomial

y2 + 6y – 18 in each of the following cases.

(a) y = 0

(b) y = 5

11. Find the value of the polynomial

y4 – 3y3 + y2 – 5 in each of the following

cases.

(a) y = 2

(b) y = –1

○○○○→→→→ Ex 2C 10, 11

Use the method of substitution.

56

�� ‘Explain Your Answer’ Questions

12. Consider the polynomial –x3 – 6x2 + 5x + 3. Amy claims that the coefficient of the x2 term is

greater than the constant term. Do you agree? Explain your answer.

The coefficient of the x2 term =

The constant term =

∵ ____ (> / = / <) ____


13. Peggy claims that if 7x2y

n is a monomial of degree 6, then the value of n is 6. Do you agree?

Explan your answer.

�� Level Up Question

14. Arrange the terms of the polynomial 5a2x4 + 7ax2 – 4a3x3 in ascending powers of

(a) a, (b) x.

(a)

(b)

5a2x4 + 7ax2 – 4a3x3

5a2x4 + 7ax2 – 4a3x3

57


Level 1

1. Determine whether each of the following algebraic expressions is a polynomial.

(a) x + 3 (b) −9 (c) k

4 − 2k

(d) 6

2tx

(e) 2x2y2 − 7 −

y

x4 (f)

5

3x3 +

3

2xy + 1


Monomial Coefficient Degree

(a) −3h

(b) 2

1k5

(c) 5m2n

(d) 0.8r3s4


Polynomial Number of terms Degree

(a) 5 + 2x

(b) p3 + 12p2q + q3

(c) hk3 − h3k2

(d) 4a −

2

b −

8

5c + 3de

For each of the polynomials in Nos. 4−−−−7, write down

(a) the constant term,

(b) the term with the highest degree and its coefficient,

(c) the degree of the polynomial.

4. −4ab + 3 5. −6 + 3m3 + 5m2

6. 5p2q + 2pq2 − pq3 7. 2r3s4 + 9r2s3 − 3rs + 12

8. Arrange the terms of each of the following polynomials in ascending powers of x.

(a) 2x − 3x3 + 5 (b) 4x2 + x6 − 7x3 − 1

9. Arrange the terms of each of the following polynomials in descending powers of y.

(a) 6y3 − 3 + 8y4 (b) −2y − 6y5 + 7y2

10. Find the value of the polynomial 5x3 − 9x + 8 in each of the following cases.

Consolidation Exercise 2C ��

58

(a) x = −2 (b) x = 0 (c) x = 1

11. Write down a monomial in variables p and q only where its degree is 5 and its coefficient is −2.

12. Karen claims that if 3x − 4xy a + 2 is a binomial of degree 3, then the value of a is 1. Do you agree?


Level 2

13. Arrange the terms of the polynomial −6a3b + 5b3 − ab2 − 7a2 in ascending powers of each of the

following variables.

(a) a (b) b

14. Arrange the terms of the polynomial 10 + 3m2n4 + 2m3n − m4n3 in descending powers of each of the

following variables.

(a) m (b) n

15. Find the value of the polynomial 4p3q2 − 5p4q + 7 in each of the following cases.

(a) p = −2, q = 1 (b) p = −1, q = −4

16. Find the value of each of the following polynomials when x =

4

1 and y =

4

3− .

(a) 9x2 + y2 + 6xy (b) x3 + x2y +

3

2xy

+

27

3y

17. Consider the polynomial

3

2x4y2 +

5

4x2yn − 8, where n is a positive integer. Frank claims that if

the degree of the polynomial is 6, then the value of n must be 1, 2 or 3 only. Do you agree? Explain

your answer.

59

Consolidation Exercise 2C (Answer)

1. (a) yes (b) yes (c) no

(d) yes (e) no (f) yes

2. Monomial Coefficient Degree

(a) −3h −3 1

(b)

2

1k5

2

1 5

(c) 5m2n 5 3

(d) 0.8r3s4 0.8 7

3. Polynomial Number

of terms Degree

(a) 5 + 2x 2 1

(b) p3 + 12p2q + q3 3 3

(c) hk3 − h3k2 2 5

(d) 4a −

2

b −

8

5c + 3de 4 2

4. (a) 3 (b) −4ab, −4 (c) 2

5. (a) −6 (b) 3m3, 3 (c) 3

6. (a) 0 (b) −pq3, −1 (c) 4

7. (a) 12 (b) 2r3s4, 2 (c) 7

8. (a) 5 + 2x − 3x3 (b) −1 + 4x2 − 7x3 + x6

9. (a) 8y4 + 6y3 − 3 (b) −6y5 + 7y2 − 2y

10. (a) −14 (b) 8 (c) 4

11. −2p4q (or other reasonable answers)

12. no

13. (a) 5b3 − ab2 − 7a2 − 6a3b

(b) −7a2 − 6a3b − ab2 + 5b3

14. (a) −m4n3 + 2m3n + 3m2n4 + 10

(b) 3m2n4 − m4n3 + 2m3n + 10

15. (a) −105 (b) −37

16. (a) 0 (b) 0

17. no

60

F2A: Chapter 2D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)



(Full Solution)

Maths Corner Exercise 2D Level 1



___________ ( )




___________ ( )




___________ ( )

61

Maths Corner Exercise 2D Multiple Choice


___________ ( )



_________

62

Book 2A Lesson Worksheet 2D (Refer to §2.4)

2.4 Addition and Subtraction of Polynomials

In polynomials, like terms can be combined under addition or subtraction.

e.g. (i) 2x + 3x = (2 + 3)x

= 5x

(ii) 4y2 – 7y2 = (4 – 7)y2

= –3y2



(a) 4x – 8 + x – 1

(b) 2x2 – 5x + 9 – x2 – x – 4

Sol (a) 4x – 8 + x – 1

= 4x + x – 8 – 1 � Group like terms.

= 5x – 9 � Combine like terms.

(b) 2x2 – 5x + 9 – x2 – x – 4

= 2x2 – x2 – 5x – x + 9 – 4

= x2 – 6x + 5


(a) 5x + 7 – 2x + 3

(b) 7x2 + 4x – 3 – 3x2 + 8x

Sol (a) 5x + 7 – 2x + 3

=

(b) 7x2 + 4x – 3 – 3x2 + 8x

=

Simplify the following expressions. [Nos. 1– 2]

1. 6x2 – 5x + 1 – 4x2 + 2x

2. xy + y2 – 9xy + 3y2

○○○○→→→→ Ex 2D 5–7

xy and –9xy are like terms.

63



(a) (3x + 1) + (2x + 5)

(b) (x – 6y) + (7x – 4y)

Sol (a) (3x + 1) + (2x + 5)

= 3x + 1 + 2x + 5

= 3x + 2x + 1 + 5

= 5x + 6

Alternative

3x + 1

+) 2x + 5

5x + 6

(b) (x – 6y) + (7x – 4y)

= x – 6y + 7x – 4y

= x + 7x – 6y – 4y

= 8x – 10y

Alternative

x – 6y

+) 7x – 4y

8x – 10y


(a) (8x – 3) + (x + 5)

(b) (7x + 2y) + (4x – 3y)

Sol (a) (8x – 3) + (x + 5)

=

Alternative

8x – 3

+)

(b) (7x + 2y) + (4x – 3y)

=

Alternative

Complete the following operations. [Nos. 3–5]

3. 4a – 2c 4. 9x – 6y 5. –3u + 9v + 2w

+) 3a – 4c +) –3x + 4y +) –5u – 8v + w

○○○○→→→→ Ex 2D 1, 3(a), 4(a)


6. (3 – 2x) + (11 – 5x)

7. (4x – y) + (9x + y)

○○○○→→→→ Ex 2D 8, 10

+(a + b) = +a + b

3x + 2x = 5x 1 + 5 = 6

+(a – b) = +a – b

Arrange the like terms in the same column.

64

8. (5x + 4x2) + (3x2 – 6x)

○○○○→→→→ Ex 2D 12(a), 13(a)

9. (6a – 8b + 3c) + (2a + 4b – 10c)

○○○○→→→→ Ex 2D 14(a), 15(a)



(a) (4x – 7) – (2x + 1)

(b) (3x – 5y) – (7y – 2x)

Sol (a) (4x – 7) – (2x + 1)

= 4x – 7 – 2x – 1

= 4x – 2x – 7 – 1

= 2x – 8

Alternative

4x – 7

–) 2x + 1

2x – 8

(b) (3x – 5y) – (7y – 2x)

= 3x – 5y – 7y + 2x

= 3x + 2x – 5y – 7y

= 5x – 12y

Alternative

3x – 5y

–) –2x + 7y

5x – 12y


(a) (8y – 5) – (3y + 2)

(b) (5x – 6y) – (y – x)

Sol (a) (8y – 5) – (3y + 2)

=

Alternative

8y – 5

–)

(b) (5x – 6y) – (y – x)

=

Alternative

Complete the following operations. [Nos. 10–12]

10. 8a – 2c 11. 4x2 – 5x – 3 12. 3xy – 7x – y

–) –3a + 4c –) 3x2 – 1 –) –xy + 2x – 6y

○○○○→→→→ Ex 2D 2, 3(b), 4(b)

4x – 2x = 2x –7 – (+1) = –7 – 1 = –8

–(a + b) = –a – b

–(a – b) = –a + b

Arrange the like terms in the same column.

65


13. (x – 9) – (2x + 10)

14. (2x – 8y) – (9x – 5y)

○○○○→→→→ Ex 2D 9, 11

15. (3x2 – 6x) – (7x – 4x2)

○○○○→→→→ Ex 2D 12(b), 13(b)

16. (2a – 3b – c) – (3a – 5b + c)

○○○○→→→→ Ex 2D 14(b), 15(b)



17. (14 – 2x + 8x2) – (3x – 5x2 + 2)

18. (3 + 8x – 2x2) + (4 – 7x2 + 6x3)

19. (6a2 + 5ac – 3c2) – (9c2 + 7ac – 2a2)

66


Level 1

Complete the following operations. [Nos. 1−−−−4]

1. (a) 3a + 2(b) 2b− 4c

+) 4a − 7 +) −5b− 3c

2. (a) 12d + 3 (b) −2e− 5f

−) 4d− 6 −) −11f

3. (a) 8k2 − 5k (b) 9t2 − 1

+) − 7k2 + 3 −) 5t2 + 3t

4. (a) 3x − 4y − 2 (b) 6mn − 4n

+) 5x − y + 8 −) 10mn− 2m + n

Simplify the following polynomials. [Nos. 5−−−−6]

5. (a) 2a3 + 6a − 3a3 − 4a (b) 3x3 − 8x2 + 6x + x + 2x2 + 5

6. (a) bc − 7ab + 3ac − 3bc + 9ab (b) 3pq + 6pr − 4qr + 2pr − 2pq + pr


7. (a) (3x − 6) + (4x + 1) (b) (8b + 3) + (2b − 9)

8. (a) (2h − 1) − (7h + 2) (b) (k + 6) − (8k − 7)

9. (a) (9x − 2y) + (5x − y) (b) (2w − 3z) + (−6w + 5z)

10. (a) (4p − 6q) − (−p + 8q) (b) (5m + n) − (−2m − 3n)

11. (a) (2c2 − 5) + (−5 − 2c2) (b) (3d − 4d 2) − (d

2 + 7d )

12. (a) (3u − v + 2) + (u − 3v − 4) (b) (2d + 5e − 6f ) − (4d − 6f − 5e)

13. (a) (−2p + 7qr) + (6p − 2qr) (b) (−xy − 4xz) − (−8yz + 3xy)

Consolidation Exercise 2D ��

67

14. (a) (1 + z) + (3z − 6) + (9 + 5z) (b) (m + 4n) − (6m − 3n) − (−3n − 8m)

Level 2

15. Complete the following operations.

(a) 2a5 − 6a3 + 8a2 + 7a (b) −4a3 + 3a2 + 2a + 8

+) 3a5 − 2a4 + 4a3 − 3a −) 9a4 − 3a3 − 5a − 1


16. (a) (m3 + 3m − 5) + (4m3 − 6m − 7) (b) (2n3 − 3n2 − 5n + 8) + (2n − 4n3 − 3 + 6n2)

17. (a) (4x3 − 3x) − (x2 − 2x3 + 3) (b) (3y − 1 − y3) − (−5 − 3y + 2y2 + 6y3)

18. (a) (3x + 5y − z) − (2y − 6x + 9z) + (−4z + 8y − 7x)

(b) (5t3 − t + 2) + (8 − 3t3 + 4t2) − (t4 − 2t2 + 6t)

19. (a) Simplify (3x2 − 5x − 4) + (2x + 9 − 2x2).

(b) Hence, find the value of the expression in (a) when x = 2.

20. (a) Simplify (4m2 + 7mn − 3n2) − (mn + 3m2 + 8n2).

(b) Hence, find the value of the expression in (a) when m = −5 and n = 3.

21. The figure on the right shows a triangle PQR.

(a) Express the perimeter of △PQR in terms of x.

(b) If x = 6, find the perimeter of △PQR.

22. There were k books in a shop. Yesterday, (3k − 18) books were sold and the shopkeeper bought (5k + 2)

new books. Today, (12 − k) books are sold. Express, in terms of k,

(a) the total number of books sold yesterday and today,

(b) the number of books in the shop at the end of today.

23. Peter has $(4x2 + 2) and Tom has $(9x − 7 − x2).

(a) How much do they have altogether?

(b) Peter and Tom want to buy a toy together but the total amount they have is not enough. If the

price of the toy is $(5x2 − 4x + 3), how much more do they need?

(Express the answers in terms of x.)

68

Consolidation Exercise 2D (Answer)

1. (a) 7a − 5

(b) −3b − 7c

2. (a) 8d + 9

(b) −2e + 6f

3. (a) k2 − 5k + 3

(b) 4t 2 − 3t − 1

4. (a) 8x − 5y + 6

(b) −4mn + 2m − 5n

5. (a) −a3 + 2a

(b) 3x3 − 6x2 + 7x + 5

6. (a) 2ab + 3ac − 2bc

(b) pq + 9pr − 4qr

7. (a) 7x − 5

(b) 10b − 6

8. (a) −5h − 3

(b) −7k + 13

9. (a) 14x − 3y

(b) −4w + 2z

10. (a) 5p − 14q

(b) 7m + 4n

11. (a) −10

(b) −5d 2 − 4d

12. (a) 4u − 4v − 2

(b) −2d + 10e

13. (a) 4p + 5qr

(b) −4xy − 4xz + 8yz

14. (a) 4 + 9z

(b) 3m + 10n

15. (a) 5a5 − 2a4 − 2a3 + 8a2 + 4a

(b) −9a4 − a3 + 3a2 + 7a + 9

16. (a) 5m3 − 3m − 12

(b) −2n3 +3n2 − 3n + 5

17. (a) 6x3 − x2 − 3x − 3

(b) −7y3 − 2y2 + 6y + 4

18. (a) 2x + 11y − 14z

(b) −t 4 + 2t3 + 6t

2 −7t + 10

19. (a) x2 − 3x + 5

(b) 3

20. (a) m2 + 6mn − 11n2

(b) −164

21. (a) (x2 + 8x − 14) m

(b) 70 m

22. (a) 2k − 6

(b) 4k + 8

23. (a) $(3x2 + 9x − 5)

(b) $(2x2 − 13x + 8)

69

F2A: Chapter 2E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)

Book Example 20


(Video Teaching)



(Full Solution)

Maths Corner Exercise 2E Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 2E Multiple Choice


___________ ( )


○ Complete and Checked ○ Problems encountered

Mark:

70

○ Skipped _________

71

Book 2A Lesson Worksheet 2E (Refer to §2.5)

2.5 Multiplication of Polynomials

(A) Method of expansion

Distributive law of multiplication:

x(a + b) = xa + xb

(a + b)x = ax + bx


Expand the following expressions.

(a) 5(6 + x)

(b) (3x – 2)(– 4x)

Sol (a) 5(6 + x)

= 5(6) + 5(x)

= 30 + 5x

(b) (3x – 2)(– 4x)

= 3x(– 4x) – 2(– 4x)

= – 12x2 + 8x


(a) 7(a – 3)

(b) (2a + 1)(–6a)

Sol (a) 7(a – 3)

=

(b) (2a + 1)(–6a)

=

Expand the following expressions. [Nos. 1–4]

1. –8(4 – 2y)

2. 2a(a – 4)

3. (– 4x + 3)(5y)

4. (6f – 5g)(–3f )

○○○○→→→→ Ex 2E 1−5

Alternatively,

ax +bx

With the table above, we have:

(a + b)x = ax + bx

x

a +b

72

Product of two binomials:

(a + b)(c + d) = (a + b)c + (a + b)d

= ac + bc + ad + bd



(a) (x – 2)(x + 3)

(b) (x – 4)(x – 5)

Sol (a) (x – 2)(x + 3)

= (x – 2)x + (x – 2)(3)

= x2 – 2x + 3x – 6

= x2 + x – 6

(b) (x – 4)(x – 5)

= (x – 4)x – (x – 4)(5)

= x2 – 4x – 5x + 20

= x2 – 9x + 20


(a) (y + 2)(y – 6)

(b) (y – 3)(y – 9)

Sol (a) (y + 2)(y – 6)

= ( )( ) – ( )( )

=

(b) (y – 3)(y – 9)

=


5. (x + 3)(x + 1)

6. (x – 7)2

7. (2 + a)(8 + a)

○○○○→→→→ Ex 2E 6−9

8. (x – 6)(2x + 6)

9. (2p – 1)(4p – 5)

10. (3r + 8)(3r – 2)

○○○○→→→→ Ex 2E 10−19

Alternatively,

x2 – 4x

–5x +20

i.e. (x – 4)(x – 5)

= x2 – 4x – 5x + 20

x – 4

x

–5

Alternatively,

ac +bc

+ad +bd

i.e. (a + b)(c + d)

= ac + bc + ad + bd

c

a +b

+d

Alternatively,

y ( )

y

( )

(x – 7)2 = (x – 7)(x – 7)

73

(B) Long multiplication


Expand the following expressions by long

multiplication.

(a) (a + 2)(a + 5)

(b) (2x + 3)(x – 5)

Sol (a) a + 2

×) a + 5

a2 + 2a

+) 5a + 10

a2 + 7a + 10

(b) 2x + 3

×) x – 5

2x2 + 3x

+) – 10x – 15

2x2 – 7x – 15

Expand the following expressions by long

multiplication.

(a) (y – 3)(y + 4)

(b) (2r – 5)(r – 2)

Sol (a) y – 3

×) y + 4

+)

(b)

×)

+)

Expand the following expressions by long multiplication. [Nos. 11–14]

11. (x – 4)(x – 7)

x – 4

×) x – 7

+)

12. (x – 1)(5x + 1)

13. (6x – 2)(x + 1)

14. (2x – 5)(3x – 7)

○○○○→→→→ Ex 2E 20−25

� Arrange the like terms in the same column.

� (a + 2)a

� (a + 2)(+5)

74

Expand and simplify the following expressions. [Nos. 15–16]

15. 5x(7 – 3x) + 6x

16. –3x(2x + 5) – (x – 8)

○○○○→→→→ Ex 2E 26, 27



17. (a) 3xy(2x + 4)

(b) (a2 – 2ab)(–5ab)

18. (a) (3x – 2y)(x + y)

(b) (x + 1)(x2 + 2x – 1)

75


Level 1

Expand the following expressions by the method of expansion. [Nos. 1−−−−4]

1. (a) 9(m + 5) (b) (2n − 3)(−4)

2. (a) a(a + 2) (b) −b(1 − 6b)

3. (a) 2x(y − 1) (b) −3y(4x + 7z)

4. (a) (h − 5g)(6h) (b) (2k − 3n)(−k3)


5. (a + 3)(a + 5) 6. (b − 4)(b + 1)

7. (x + 2)2 8. (9 + y)(y − 9)

9. (3m + 4)(m + 6) 10. (−5 + a)(5a − 1)

11. (−6 − n)(2 − 5n) 12. (4 − 7b)(8 − 3b)

13. (3r2 + 2)(r − 1) 14. (2 − 3s2)(1 + 4s)

Expand the following expressions by long multiplication. [Nos. 15−−−−20]

15. (a + 8)(a + 1) 16. (b + 1)(b − 9)

17. (k − 4)(k − 5) 18. (h − 2)(−h + 5)

19. (2m + 7)(m − 4) 20. (2n − 3)(3n + 1)

Expand and simplify the following expressions. [Nos. 21−−−−22]

21. 12p + p(p − 5)

22. 3(r − 4) − (5r − 7)r

Level 2

Consolidation Exercise 2E ��

76


23. (a) 5xy(2x − 3y) (b) (x2 − 2xy)(−4xy)

24. (a) −4(m2 − 3m − 5) (b) (1 − 3n − 2n2)(−3n)

25. (a) (h + k)(k − h) (b) (m + 2p)(−9m + 4p)

26. (a) (n − 3q)(2n − 5q) (b) (3r + 5s)(6r − 7s)

27. (a) (y − 1)(y2 + y + 1) (b) (x2 − 2x + 4)(x + 2)

28. (a) (3m + 1)(m2 − 5m + 6) (b) (2n2 + 8n − 3)(4 − n)

Expand the following expressions by long multiplication. [Nos. 29−−−−30]

29. (a) (a + 2b)(3a + 8b) (b) (−4p + 7q)(5p − 3q)

30. (a) (m2 + 6m + 9)(m − 3) (b) (3n − 2)(−4 + 2n − n2)

31. (a) Expand 6x(2 − x2).

(b) Hence, simplify (2x2 + 12x + 8) − 6x(2 − x2).

Expand and simplify the following expressions. [Nos. 32−−−−35]

32. 2a(a − 1) + 3a(2a − 3) 33. −(n + 4)(3n − 2) − (8 − 3n2)

34. (4b − 1)(2b − 5) − (5 − b2 − 7b) 35. (m + 2)(m − 4)(4m − 1)

36. A factory produced (9r − 2) identical bricks last week. If the weight of each brick is (2r + 5) kg,

express the total weight of the bricks in terms of r and arrange the terms in descending powers of r.

37. The figure on the right shows a pentagon PQRST.

(a) Express the area of PQRST in terms of x.

(b) If x = 2, find the area of PQRST.

77

Consolidation Exercise 2E (Answer)

1. (a) 9m + 45 (b) −8n + 12

2. (a) a2 + 2a (b) −b + 6b2

3. (a) 2xy − 2x (b) −12xy − 21yz

4. (a) 6h2 − 30gh (b) −2k4 + 3nk3

5. a2 + 8a + 15

6. b2 − 3b − 4

7. x2 + 4x + 4

8. y2 − 81

9. 3m2 + 22m + 24

10. 5a2 − 26a + 5

11. 5n2 + 28n − 12

12. 21b2 − 68b + 32

13. 3r3 − 3r2 + 2r − 2

14. 2 + 8s − 3s2 − 12s3

15. a2 + 9a + 8

16. b2 − 8b − 9

17. k2 − 9k + 20

18. −h2 + 7h − 10

19. 2m2 − m − 28

20. 6n2 − 7n − 3

21. p2 + 7p

22. −5r2 + 10r − 12

23. (a) 10x2y − 15xy2

(b) −4x3y + 8x2y2

24. (a) −4m2 + 12m + 20

(b) −3n + 9n2 + 6n3

25. (a) k2 − h2

(b) −9m2 −14mp + 8p2

26. (a) 2n2 − 11nq + 15q2

(b) 18r2 + 9rs − 35s2

27. (a) y3 − 1

(b) x3 + 8

28. (a) 3m3 − 14m2 + 13m + 6

(b) −2n3 + 35n − 12

29. (a) 3a2 + 14ab + 16b2

(b) −20p2 + 47pq −21q2

30. (a) m3 + 3m2 − 9m − 27

(b) −3n3 + 8n2 − 16n + 8

31. (a) 12x − 6x3

(b) 6x3 + 2x2 + 8

32. 8a2 − 11a

33. −10n

34. 9b2 −15b

35. 4m3 − 9m2 − 30m + 8

36. (18r2 + 41r − 10) kg

37. (a)

−+ 30

2

2310 2

xx cm2

(b) 33 cm2

78

F2A: Chapter 2F

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 21


(Video Teaching)

Book Example 22


(Video Teaching)

Book Example 23


(Video Teaching)

Book Example 24


(Video Teaching)

Book Example 25


(Video Teaching)

Book Example 26


(Video Teaching)

Book Example 27


(Video Teaching)

79



(Full Solution)

Maths Corner Exercise 2F Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 2F Multiple Choice


___________ ( )



_________

80

Book 2A Lesson Worksheet 2F (Refer to §2.6)

2.6 Factorization of Polynomials

Factorization is the process of converting a polynomial into the product of its factors.

e.g. Consider the relation between x(x + 3) and x2 + 3x.

2.6A Taking out Common Factors

ab + ac = a(b + c)


Factorize the following expressions.

(a) 5a + 5b

(b) –7x – 21

(c) 3ah – 3ak

Sol (a) 5a + 5b

= 5(a + b)

(b) –7x – 21

= –7x – 7(3)

= –7(x + 3)

(c) 3ah – 3ak

= 3a(h – k)

Factorize the following expressions.

(a) 8x + 8y

(b) –12 – 2a

(c) 4ax – 4cx

Sol (a) 8x + 8y

=

(b) –12 – 2a

= ( )( ) – 2a

=

(c) 4ax – 4cx

=

Factorize the following expressions. [Nos. 1–10]

1. xy + 7x

2. –3d – 15

○○○○→→→→ Ex 2F 1−4

Expansion

Factorization

x(x + 3) x2 + 3x

Factors of 7x: 7, x Factors of 21: 1, 3, 7, 21 ∴ The common factor

is 7.

Check the answer: 5(a + b) = 5a + 5b

Factors of 3ah: 3, a, h

Factors of 3ak: 3, a, k ∴ The common factor

is 3 × a (i.e. 3a).

81

3. 16e2 + 8e

4. 15f 2 – 12ef

5. –gh2 – 2gh

6. 5x3 – 20x2

○○○○→→→→ Ex 2F 5−10

7. 2x + xy – 4xz

8. 3h2k + 2h2 – 5h2k2

○○○○→→→→ Ex 2F 11−14

9. (p + 4)q + (p + 4)r

10. 9x(3x – 2y) – 2(3x – 2y)

○○○○→→→→ Ex 2F 15−17

e2 = e ⋅ e

The common factor

is ( ).

82

2.6B Grouping Terms


Factorize 3a + 3b + ac + bc.

Sol 3a + 3b + ac + bc

= 3(a + b) + c(a + b)

= (a + b)(3 + c)

Factorize ax + ay + 2x + 2y.

Sol ax + ay + 2x + 2y

= a( ) + 2( )

= ( )( )

Factorize the following expressions. [Nos. 11–18]

11. m – n + rm – rn

12. 3x2 – 2x + 3xy – 2y

13. ac + ab – 5c – 5b

14. x + x2 – r – rx

15. x – y – ax + ay

16. ar – 2a – 3r + 6

○○○○→→→→ Ex 2F 18−26

17. 8u – 8v + cv – cu

18. rs – rk – ak + as

○○○○→→→→ Ex 2F 27, 28

� Step 1111:

Group the terms.

Step 2222:

Take out the common

factor in each group.

Step 3333:

Take out the common factor (i.e. a + b) of the groups.

� Step 1111

� Step 2222

� Step 3333

–5c – 5b = –5(c b)

–ax + ay = –a(x y)

Rearrange the terms if necessary.

�

�

rm – rn = r(m n)

83


19. Sofia claims that the two expressions xy + y – x – 1 and xy – y + x – 1 have the common factor

x – 1. Do you agree? Explain your answer.

xy + y – x – 1 =

xy – y + x – 1 =

∵ The two expressions (have / do not have) the common factor x – 1.



20. Factorize 2ac – 8bc + 10c2.

21. Factorize 9m(x + y) – 6n(x + y).

22. Factorize 3xy – 9y + 2x – 6.

23. Factorize ax + y + x + ay.

ax + y + x + ay = ax + ay + __________

=

Rearrange the terms for grouping.

84


Level 1

Factorize the following expressions. [Nos. 1−−−−28]

1. 6a − 6b 2. 7c + 14

3. mn + m 4. 5q − pq

5. 3rs + 9rt 6. −2uv − 2vw

7. 4x2 − 8x 8. 9h2 + 12hk

9. 7a3 + a4 10. −m2n3 − m3n2

11. 8x + 4y − 12 12. 3d − de − df

13. m2 − 2mn − 3m 14. 3ab3 − 5a2b2 + b2

15. p(w + z) − q(w + z) 16. (7m + 9) + n(9 + 7m)

17. 2b(x − y) + a(x − y) 18. uv − 11u + v − 11

19. kr + ks + s + r 20. a − b + ac − bc

21. h + k − hx − kx 22. wz + z + 6w + 6

23. pn − qn − 4p + 4q 24. 5x + 15 − xy − 3y

25. ab − 7b − ac + 7c 26. m2n + m2 − 2n − 2

27. hx − hy − 4x + 4y 28. 3y − 8xy − 8xz + 3z

Consolidation Exercise 2F ��

85

Level 2

Factorize the following expressions. [Nos. 29−−−−44]

29. 4pq − 12pr + 20ps

30. −9ab − 18a2 − 3a

31. 15m2 + 10m3 − 25m5

32. 8r2s2 − 20r2s3 + 24r3s2t

33. 6x(2a − b) + 3y(2a − b)

34. (3k + 5)(m + n) − 4(n + m)

35. (3b − 7)(h + 2k) − 2b(h + 2k)

36. (m − n)(p + 2q) + (n − m)(3p − q)

37. 4m − nk + 4n − mk

38. 10pr − 4qs + 8pq − 5rs

39. 8tu + t − 4t2 − 2u

40. −4nk − n2 − nh − 4hk

41. 4pm − 4qm − 4rm − pn + qn + rn

42. ax − bx + 2cx − 2cy + by − ay

43. 4x − 6 + (2x − 3)2

44. (y + 7)(y − 7) + 21 − 3y

86

Consolidation Exercise 2F (Answer)

1. 6(a − b)

2. 7(c + 2)

3. m(n + 1)

4. q(5 − p)

5. 3r(s + 3t)

6. −2v(u + w)

7. 4x(x − 2)

8. 3h(3h + 4k)

9. a3(7 + a)

10. −m2n2(n + m)

11. 4(2x + y − 3)

12. d(3 − e − f )

13. m(m − 2n − 3)

14. b2(3ab − 5a2 + 1)

15. (w + z)(p − q)

16. (7m + 9)(1 + n)

17. (x − y)(2b + a)

18. (v − 11)(u + 1)

19. (r + s)(k + 1)

20. (a − b)(1 + c)

21. (h + k)(1 − x)

22. (w + 1)(z + 6)

23. (p − q)(n − 4)

24. (x + 3)(5 − y)

25. (a − 7)(b − c)

26. (n + 1)(m2 − 2)

27. (x − y)(h − 4)

28. (3 − 8x)(y + z)

29. 4p(q − 3r + 5s)

30. −3a(3b + 6a + 1)

31. 5m2(3 + 2m − 5m3)

32. 4r2s2(2 − 5s + 6rt)

33. 3(2a − b)(2x + y)

34. (m + n)(3k + 1)

35. (h + 2k)(b − 7)

36. (m − n)(3q − 2p)

37. (m + n)(4 − k)

38. (2p − s)(5r + 4q)

39. (4t − 1)(2u − t)

40. −(4k + n)(n + h)

41. (p − q − r)(4m − n)

42. (a − b + 2c)(x − y)

43. (2x − 3)(2x − 1)

44. (y − 7)(y + 4)

87

F2A: Chapter 3A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

88



___________ ( )



_________

89

Book 2A Lesson Worksheet 3A (Refer to §3.1)

3.1A Meaning of Identities and the Tests for Identities

(a) For an equation in x, if ANY value of x can satisfy the

equation, then the equation is called an identity.

(b) For an identity, the like terms on both sides must be the same.

e.g. 2x ≡ x + x is an identity.

3x = 1 is not an identity.


Prove that the equation x + 4x = 5x is an

identity.

Sol

L.H.S. = x + 4x

= 5x

R.H.S. = 5x

∵ L.H.S. = R.H.S.

∴ x + 4x ≡ 5x

Prove that the equation –6y = 2y – 8y is an

identity.

Sol L.H.S. = –6y

R.H.S. =

∵ L.H.S.

∴

1. Prove that the equation 2(x – 1) = 2x – 2 is

an identity.

L.H.S. =

2. Prove that the equation

–12y + 8 = – 4(3y – 2) is an identity.

○○○○→→→→ Ex 3A 1– 4

� To indicate an identity, we

use the identity symbol ‘≡’ to replace ‘=’.

Step 1111: Simplify L.H.S. and/or R.H.S.

Step 2222: Check if L.H.S. = R.H.S.

2(x – 1)

90


Prove that the equation 3(2x + 1) – x = 5x + 7

is not an identity.

Sol Method 1

L.H.S. = 3(2x + 1) – x

= 6x + 3 – x

= 5x + 3

R.H.S. = 5x + 7

∵ L.H.S. ≠ R.H.S.

∴ 3(2x + 1) – x = 5x + 7 is not an

identity.

Method 2

When x = 0,

L.H.S. = 3[2(0) + 1] – 0 = 3

R.H.S. = 5(0) + 7 = 7

∵ L.H.S. ≠ R.H.S.

∴ 3(2x + 1) – x = 5x + 7 is not an

identity.

Prove that the equation 4x + 6 = 2(2x – 5) + 4

is not an identity.

Sol Method 1

L.H.S. = 4x + 6

R.H.S. =

∵ L.H.S.

∴

Method 2

When x = ,

L.H.S. =

R.H.S. =

∵ L.H.S.

∴

3. Prove that the equation

3(5x + 2) = 6(2x + 1) is not an identity.

Method 1

L.H.S. =

R.H.S. =

Method 2

When x = ,

L.H.S. =

R.H.S. =

4. Prove that the equation y2 – 4 = (y – 2)2 is

not an identity.

Method 1

L.H.S. =

R.H.S. = (y – 2)2

= (y – 2)( )

=

Method 2

○○○○→→→→ Ex 3A 5–8

Find ONE value of x such that

L.H.S. ≠ R.H.S.

91

Determine whether each of the following equations is an identity. [Nos. 5–6]

5. 5(2x + 4) = 10(1 + 2x)

L.H.S. =

R.H.S. =

∵ L.H.S. (= / ≠) R.H.S.

∴

6. (x + 3)(x – 2) = x2 + x – 6

L.H.S. = (x + 3)(x – 2)

= (x + 3)( ) – (x + 3)( )

=

○○○○→→→→ Ex 3A 9–14

3.1B Finding Unknown Constants in an Identity

Method 1:

Consider an identity in x with unknown constant(s).

Step 1111: Expand and simplify the two sides of the identity.

Step 2222: Compare the like terms on both sides.

e.g. Consider Px + Q ≡ 6x – 4, where P and Q are constants.

Comparing like terms on both sides, we have

P = 6

Q = – 4


If 4(x + 5) ≡ Ax + B, where A and B are

constants, find A and B.

Sol L.H.S. = 4(x + 5)

= 4x + 20

∴ 4x + 20 ≡ Ax + B

Comparing like terms on both sides,

we have

A = 4

B = 20

If Px2 + Qx ≡ 7x(x – 2), where P and Q are

constants, find P and Q.

Sol R.H.S. = 7x(x – 2)

=

∴

Comparing like terms on both sides,

we have

P =

Q =

� P x + Q

≡ 6 x – 4

92

Find the unknown constants (denoted by capital letters) in each of the following identities.

[Nos. 7–8]

7. 3(y + A) ≡ By – 3

8. 5x(x + 2) + Px ≡ Qx2 + 8x

○○○○→→→→ Ex 3A 15–20

Method 2:

Consider an identity in x with unknown constant(s).

Substitute a suitable value of x into the identity to find each

unknown constant.


If (x – 2)(x + 1) ≡ x2 + Ax + B, where A and B

are constants, find A and B.

Sol For the identity

(x – 2)(x + 1) ≡ x2 + Ax + B,

when x = 0,

(0 – 2)(0 + 1) = 02 + A(0) + B

∴ B = –2

when x = 2,

(2 – 2)(2 + 1) = 22 + A(2) – 2

0 = 4 + 2A – 2

2A = –2

∴ A = –1

If (x – 1)(x – 3) ≡ x2 + Px + Q, where P and Q

are constants, find P and Q.

Sol For the identity

(x – 1)(x – 3) ≡ x2 + Px + Q,

when x = ,

( )( ) = ( )2 + P( ) + Q

∴ =

when x = ,

Remember to combine like

terms.

93


[Nos. 9–10]

9. (x + 4)(x – 2) ≡ x2 + Ax + B

10. (4y – 3)(y + 1) ≡ 4y2 + Py + Q

○○○○→→→→ Ex 3A 21, 22


11. Determine whether the equation 5(3x + 1) + 3(x – 4) = 6(3x – 1) is an identity.

12. If (x + 2)(x – A) ≡ x2 + Bx – 20, where A and B are constants, find A and B.

94

3 Identities

Level 1

Prove that each of the following equations is an identity. [Nos. 1–4]

1. 6(y − 2) = 6y − 12 2. 4(2x + 3) = 2(4x + 6)

3. 9 − y2 = (3 + y)(3 − y) 4. x(x − 5) + 8x = x2 + 3x

Prove that each of the following equations is not an identity. [Nos. 5–8]

5. 5x − 2 = 5(x − 1) + 4 6. 3(5x + 2) = 5(3x + 1) + x

7. y(2 − y) + 5 = y2 − 2y + 5 8. (y − 4)2 = y2 − 16


9. 2x + 5x = 11x − 4x 10. −4(2y + 3) = −(8y − 12)

11. 3(x + 2) − 5x = −2(x − 3) 12. 4y(4 − 2y) = 2y(4y − 7) − 2y

13. (x − 5)2 − 3x = x2 − 13x − 25 14. (y + 5)(y − 7) + 35 = y2 − 2y

Find the unknown constants P and Q in each of the following identities. [Nos. 15–24]

15. 3x − 8 ≡ Px − Q 16. 6x + 1 ≡ P + Qx

17. 5(x + P) ≡ Qx + 20 18. 4(3x − 2) + P ≡ Qx − 10

19. 7y(y + 2) ≡ Py2 + Qy 20. y(4y + P) ≡ 6y − Qy2

21. (Px − 3)x − 2 ≡ −5x2 − 3x + Q 22. x2 + Px − Q ≡ (x − 3)(x + 5)

23. Py2 + 17y + Q ≡ (y + 4)(2y + 9) 24. (2y − 1)(3y − 4) ≡ 6y2 − Py + Q

Level 2


95

Prove that each of the following equations is an identity. [Nos. 25–27]

25. (7 − 3x)2 = 9x2 − 42x + 49

26. y(y + 11) + 30 = (y + 5)(y + 6)

27. 4(7x − 6 + 3x2) = (2x + 6)(6x − 4)

Prove that each of the following equations is not an identity. [Nos. 28–30]

28. 3 − 4x(x − 2) = (3 − 2x)(2x − 1)

29. (5y − 6)(3y + 5) + 3y = 15y2 − 10y − 30

30. 6

74

6

7

2

+=

−+

nnn


31. (2p – 3)(p + 6) – 4p = (p + 2)(2p – 9) 32. 4m − 3m(2m − 5) = −(4 − 3m)(2m + 9)

33. (n – 2)2 + n(5n – 6) = 2(3n – 2)(n – 1) 34. 6

32

9

2

18

54 −=

−−

− xxx


[Nos. 35–42]

35. (x + 3)(x + A) ≡ x2 + Bx − 6 36. (Ax − 3)(x − 2) ≡ 5x2 + Bx + 6

37. (Ax − B)(3x + 2) ≡ −6x2 − 19x − 10 38. (3x + 1)2 − A ≡ 9x2 − Bx − 3

39. 4(x + 5) + 2(2x – 5) ≡ Ax2 + Bx + C 40. (8x + 2)(Ax – B) ≡ 24x2 + Cx – 8

41. x(Ax − 4) − B ≡ (2x + C)2 + 2 42. (x − 4)(Ax − 1) ≡ 6x2 + B(5x − 1) + C

43. It is given that (x + A)(x + B) ≡ x2 + x + C, where A, B and C are constants.

(a) Express B in terms of A. Hence, express C in terms of A.

(b) Write down two sets of possible values of A, B and C.

96


9. yes 10. no

11. yes 12. no

13. no 14. yes

15. P = 3, Q = 8 16. P = 1, Q = 6

17. P = 4, Q = 5 18. P = −2, Q = 12

19. P = 7, Q = 14 20. P = 6, Q = −4

21. P = −5, Q = −2 22. P = 2, Q = 15

23. P = 2, Q = 36 24. P = 11, Q = 4

31. no 32. no

33. yes 34. yes

35. A = −2, B = 1 36. A = 5, B = −13

37. A = −2, B = 5 38. A = 4, B = −6

39. A = 0, B = 8, C = 10

40. A = 3, B = 4, C = −26

41. A = 4, B = −3, C = −1

42. A = 6, B = −5, C = −1

43. (a) B = 1 – A, C = A – A2

(b) A = 1, B = 0, C = 0;

A = 2, B = –1, C = –2


97

F2A: Chapter 3B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )


○ Complete and Checked ○ Problems encountered

Mark:

98

○ Skipped _________

99


3.2 The Difference of Two Squares Identity

a2 – b2 ≡ (a + b)(a – b)


Expand

(a) (x + 4)(x – 4) , (b) (3 – y)(3 + y).

Sol (a) (x + 4)(x – 4)

= x2 – 42

= x2 – 16

(b) (3 – y)(3 + y)

= 32 – y2

= 9 – y2

Expand

(a) (y – 7)(y + 7) , (b) (8 + x)(8 – x).

Sol (a) (y – 7)(y + 7)

= ( )2 – ( )2

=

(b) (8 + x)(8 – x)

= ( )2 – ( )2

=

1. Expand (10 + x)(10 – x).

2. Expand (–2 + y)(–2 – y).

○○○○→→→→ Ex 3B 1– 4


Expand (2n – 5)(2n + 5).

Sol (2n – 5)(2n + 5)

= (2n)2 – 52

= 4n2 – 25 � (2n)2 = 22n2

Expand (9x + 1)(9x – 1).

Sol (9x + 1)(9x – 1)

= ( )2 – ( )2

=


3. (4a + 3)(4a – 3)

4. (5 – 7h)(5 + 7h)

○○○○→→→→ Ex 3B 5–7

� We can also write it as

(a + b)(a – b) ≡ a2 – b2.

a = b =

a = x, b = 4

a = 3, b = y a = b =

100

5. (1 + 6x)(6x – 1)

(1 + 6x)(6x – 1)

= (6x + )(6x – 1)

=

6. (–5y + 2)(5y + 2)

○○○○→→→→ Ex 3B 8, 9

7. 2(x + 7)(x – 7)

2(x + 7)(x – 7)

= 2[( )2 – ( )2]

=

8. –5(3k + 2)(3k – 2)

○○○○→→→→ Ex 3B 10–15




(a) 782 – 222

(b) 41 × 39

Sol (a) 782 – 222

= (78 + 22)(78 – 22) �

= 100 × 56

= 5 600

(b) 41 × 39

= (40 + 1)(40 – 1)

= 402 – 12 �

= 1 600 – 1

= 1 599



(a) 552 – 452

(b) 28 × 32

Sol (a) 552 – 452

= ( + )( – )

= ( ) × ( )

=

(b) 28 × 32

= ( – )( + )

= ( )2 – ( )2

=


9. 862 – 142

10. 322 – 222

○○○○→→→→ Ex 3B 16–19

a2 – b2

≡ (a + b)(a – b)

(a + b)(a – b)

≡ a2 – b2

101

11. 59 × 61

12. 103 × 97

○○○○→→→→ Ex 3B 20, 21


13. Expand (5x – 9y)(5x + 9y).

14. Expand (11 + m2)(11 – m2).

15. Without using a calculator, find the value of 3052 – 2952.

(m2)2 = m2 × 2 = m( )

102

3 Identities

Level 1


1. (p + 1)(p − 1) 2. (q + 6)(q − 6)

3. (7 − r)(7 + r) 4. (−4 − m)(−4 + m)

5. (5n + 1)(5n − 1) 6. (2n + 5)(2n − 5)

7. (−6 + 3x)(−6 − 3x) 8. (7y − 8)(8 + 7y)

9. (−9 + z)(9 + z) 10. (–5x + 3)(5x + 3)

11. 8(a + 3)(a − 3) 12. −5(b − 4)(b + 4)

13. 4(5c + 2)(5c − 2) 14. 2(4k − 1)(4k + 1)

15. −9(3 + 2t)(3 − 2t) 16. 6(−4 + 3x)(−4 − 3x)

Without using a calculator, use the difference of two squares identity to find the values of the following

expressions. [Nos. 17–22]

17. 812 − 192

18. 792 − 692

19. 1042 − 962

20. 1622 − 1382

21. 53 × 47

22. 95 × 105

Level 2



103

23. (2a + 5b)(2a − 5b) 24. (4c − 3d)(−4c − 3d)

25.

+

−

22

pqq

p 26.

−

+ cdcd 2

5

12

5

1

27. 8(3m – 2n)(3m + 2n) 28. −2(5x + 6y)(6y − 5x)

29. 9

−

+

9393

nmnm 30. (k2 + 8)(k2 − 8)

31. (4t2 − 7)(7 + 4t2) 32. (ab − 5)(−5 − ab)

33. (3m + 9n)(m − 3n) 34.

−

+

842

xyy

x

35. (a) Expand (5a + 3b)(5a − 3b).

(b) Hence, expand 15

−

+ b

aa

b

3

5

5

3.

36. (a) Expand (2x + 7y)(7y − 2x).

(b) Hence, expand

−

+ xyyx

7

2

2

7.

Without using a calculator, use the difference of two squares identity to find the values of the following

expressions. [Nos. 37–40]

37. 4 0022 − 3 9982 38. 47.52 − 2.52

39. 510 × 490 40. 293

1 × 30

3

2

41. (a) Express 497 × 503 in the form a2 − b2.

(b) Hence, find the value of 497 × 503 − 4992.

42. (a) Express 1 004 × 996 in the form a2 − b2.

(b) Hence, find the value of 1 0012 – 1 004 × 996.

104


1. p2 − 1 2. q2 − 36

3. 49 − r2 4. 16 − m2

5. 25n2 – 1 6. 4n2 − 25

7. 36 − 9x2 8. 49y2 − 64

9. z2 − 81 10. 9 – 25x2

11. 8a2 − 72 12. 80 − 5b2

13. 100c2 − 16 14. 32k2 − 2

15. 36t2 − 81 16. 96 − 54x2

17. 6 200 18. 1 480

19. 1 600 20. 7 200

21. 2 491 22. 9 975

23. 4a2 − 25b2 24. 9d 2 − 16c2

25. 4

2p

− q2 26. 25

1 – 4c2d

2

27. 72m2 – 32n2 28. 50x2 – 72y2

29. m2 –

9

2n

30. k 4 − 64

31. 16t 4 − 49 32. 25 − a2b2

33. 3m2 − 27n2 34. 4

2y

−

16

2x

35. (a) 25a2 − 9b2 (b) 25a2 − 9b2

36. (a) 49y2 − 4x2 (b) 2

7y2 −

7

2x2

37. 32 000 38. 2 250

39. 249 900 40. 8999

5

41. (a) 5002 – 32 (b) 990

42. (a) 1 0002 – 42 (b) 2 017

105

F2A: Chapter 3C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

106

Maths Corner Exercise 3C Multiple Choice


___________ ( )



_________

107

Book 2A Lesson Worksheet 3C (Refer to §3.3)

3.3A Square of the Sum of Two Numbers

(a + b)2 ≡ a2 + 2ab + b2


Expand

(a) (x + 1)2, (b) (4 + y)2.

Sol (a) (x + 1)2

= x2 + 2(x)(1) + 12

= x2 + 2x + 1

(b) (4 + y)2

= 42 + 2(4)(y) + y2

= 16 + 8y + y2

Expand

(a) (x + 6)2, (b) (3 + y)2.

Sol (a) (x + 6)2

= ( )2 + 2( )( ) + ( )2

=

(b) (3 + y)2

= ( )2 + 2( )( ) + ( )2

=

1. Expand the following expressions.

(a) (x + 2)2 = ( )2 + 2( )( ) + ( )2

=

(b) (x + 3)2 =

(c) (x + 5)2 =

(d) (x + 7)2 =

(e) (x + 9)2 =

○○○○→→→→ Ex 3C 1, 2, 5


Expand (5y + 3)2.

Sol (5y + 3)2 = (5y)2 + 2(5y)(3) + 32

= 25y2 + 30y + 9

Expand (4 + 9x)2.

Sol (4 + 9x)2 = ( )2 + 2( )( ) + ( )2

=

2. Expand (10a + 1)2.

3. Expand (–3 + 8c)2.

○○○○→→→→ Ex 3C 7, 8, 11, 14

a = b =

a = b =

Try to memorize the following square numbers.

12 = 1 52 = 25 92 = 81 22 = 4 62 = 36 102 = 100 32 = 9 72 = 49 112 = 121

42 = 16 82 = 64 122 = 144

a = x, b = 1

a = 4, b = y

108

3.3B Square of the Difference of Two Numbers

(a – b)2 ≡ a2 – 2ab + b2


Expand

(a) (x – 1)2,

(b) (5 – y)2.

Sol (a) (x – 1)2

= x2 – 2(x)(1) + 12

= x2 – 2x + 1

(b) (5 – y)2

= 52 – 2(5)(y) + y2

= 25 – 10y + y2

Expand

(a) (y – 7)2,

(b) (3 – x)2.

Sol (a) (y – 7)2

= ( )2 – 2( )( ) + ( )2

=

(b) (3 – x)2

= ( )2 – 2( )( ) + ( )2

=

4. Expand the following expressions.

(a) (x – 2)2 = ( )2 – 2( )( ) + ( )2

=

(b) (x – 4)2 =

(c) (x – 6)2 =

(d) (x – 8)2 =

(e) (x – 10)2 =

○○○○→→→→ Ex 3C 3, 4, 6


Expand (3 – 4r)2.

Sol (3 – 4r)2 = 32 – 2(3)(4r) + (4r)2

= 9 – 24r + 16r2

Expand (6y – 5)2.

Sol (6y – 5)2 = ( )2 – 2( )( ) + ( )2

=

5. Expand (4 – 5x)2.

6. Expand (–2c – 7)2.

○○○○→→→→ Ex 3C 9, 10, 12, 13

a = x, b = 1 a = b =

a = 5, b = y a = b =

109

7. Expand 2(x + 4)2.

2(x + 4)2

= 2[( )2 + 2( )( ) + ( )2]

=

8. Expand 3(2 – 3n)2.

○○○○→→→→ Ex 3C 15–18




(a) 312

(b) 982

Sol (a) 312 = (30 + 1)2

= 302 + 2(30)(1) + 12

= 900 + 60 + 1

= 961

(b) 982 = (100 – 2)2

= 1002 – 2(100)(2) + 22

= 10 000 – 400 + 4

= 9 604



(a) 1052

(b) 492

Sol (a) 1052 = (100 + )2

= ( )2 + 2( )( ) + ( )2

=

(b) 492 = (50 – )2

= ( )2 – 2( )( ) + ( )2

=


9. 532 10. 2022

11. 392 12. 672

○○○○→→→→ Ex 3C 19–21

(a + b)2 ≡

(a – b)2 ≡

110


13. Expand (8x + 3y)2.

14. Expand (5p – 12q)2.

15. (a) Prove that the equation (m + 1)2 + (m – 1)2 = 2(m2 + 1) is an identity.

(b) Using the result of (a), find the value of 1012 + 992 without using a calculator.

111

3 Identities

Level 1


1. (m + 3)2 2. (5 + n)2 3. (p − 7)2 4. (9 − q)2

5. (−x + 6)2 6. (−1 + y)2 7. (−8 − y)2 8. (−x – 4)2

9. (3a + 2)2 10. (8b + 5)2 11. (4k − 1)2 12. (6t − 7)2

13. (2 + 9r)2 14. (5 − 3s)2 15. (−8 − 3x)2 16. (−2y + 7)2

17. 3(h – 1)2 18. 5(2 + k)2 19. 2(5x + 6)2 20. 4(3 − 4p)2

Without using a calculator, use the perfect square identities to find the values of the following expressions.

[Nos. 21–24]

21. 622 22. 972 23. 2012 24. 2982

Level 2


25. (3m + n)2 26. (5a − 2b)2 27. (−4x − 3y)2 28.

2

7

+

nm

29.

2

52

− s

r 30.

2

65

+

vu 31.

3

1(3h + 9k)2 32. −5(2x2 − 3)2

33. [4(3s − 1)]2 34. [3(x + 6y)]2 35. (8p2 − 4q)2 36. 6(mn + 2)2

37. (a) Expand (2 + m)2.

(b) Hence, expand (2 + m + 3n)2.

38. (a) Expand (y − 3z)2.

(b) Hence, expand (x + y − 3z)(x − y + 3z).

Without using a calculator, use the perfect square identities to find the values of the following expressions.

[Nos. 39–42]

39. 5982 40. 1 0032 41. 20.12 42. 79.92

Consolidation Exercise 3C ��

112

Consolidation Exercise 3C (Answer)

1. m2 + 6m + 9 2. 25 + 10n + n2

3. p2 – 14p + 49 4. 81 − 18q + q2

5. x2 − 12x + 36 6. 1 – 2y + y2

7. 64 + 16y + y2 8. x2 + 8x + 16

9. 9a2 + 12a + 4 10. 64b2 + 80b + 25

11. 16k2 − 8k + 1 12. 36t2 − 84t + 49

13. 4 + 36r + 81r2 14. 25 − 30s + 9s2

15. 64 + 48x + 9x2 16. 4y2 − 28y + 49

17. 3h2 – 6h + 3 18. 20 + 20k + 5k2

19. 50x2 + 120x + 72 20. 36 − 96p + 64p2

21. 3 844 22. 9 409

23. 40 401 24. 88 804

25. 9m2 + 6mn + n2 26. 25a2 − 20ab + 4b2

27. 16x2 + 24xy + 9y2 28. m2 +

7

2mn +

49

2n

29. 4

2r

– 5rs + 25s2 30. 25

2u

+

15

uv +

36

2v

31. 3h2 + 18hk + 27k2 32. −20x4 + 60x2 − 45

33. 144s2 − 96s + 16 34. 9x2 + 108xy + 324y2

35. 64p4 − 64p2q + 16q2

36. 6m2n2 + 24mn + 24

37. (a) 4 + 4m + m2

(b) 4 + 4m + m2 + 12n + 6mn + 9n2

38. (a) y2 − 6yz + 9z2

(b) x2 − y2 + 6yz − 9z2

39. 357 604 40. 1 006 009

41. 404.01 42. 6 384.01

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